Now You Try

<p> Find the indicated side <code class='latex inline'>x</code> or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/1515" />

<p> Use the Law of Sines to find the indicated side x or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/6852" />

<p>Calculate each unknown angle to the nearest degree and each unknown In length to the nearest tenth of a centimetre.</p><img src="/qimages/929" />

<p> Sketch the triangle and then solve the triangle using the Law of Sines.</p><p><code class='latex inline'>\angle B = 10^{\circ}</code>, <code class='latex inline'>\angle C = 100^{\circ}</code>, <code class='latex inline'>c = 115</code>.</p>

<p>Draw a labelled diagram for the triangle. Then calculate the required side length or angle measure.</p><p>In <code class='latex inline'>\triangle WXY</code>, <code class='latex inline'>w = 12.0 cm</code>, <code class='latex inline'>y = 10.5 cm</code>, and <code class='latex inline'>\angle W = 60^o</code>. Determine the measure of <code class='latex inline'>\angle Y</code>.</p>

<p> Use the Law of Sines to find the indicated side x or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/257" />

<p>Calculate each unknown angle to the nearest degree and each unknown In length to the nearest tenth of a centimetre.</p><img src="/qimages/928" />

<p>For each of the following, draw possible diagrams that match the given measurements. Then, calculate the length of side <code class='latex inline'>c</code>. If the calculation cannot be made, explain why. </p><p>In <code class='latex inline'>\triangle ABC, a = 13 cm, b = 21 cm</code>, and <code class='latex inline'>\angle A = 29^o</code>. </p>

<p>The manufacturer of a reclining lawn chair is planning to cut notches on the back of the chair so that you can recline at an angle of 30° as shown.</p><p>a) What is the measure of <code class='latex inline'>\angle B</code>, to the nearest degree, for the chair to be reclined at the proper angle?</p><p>b) Determine the distance from A to B to the nearest centimetre.</p><img src="/qimages/5300" />

<p>Solve for the unknown side length or angle measure set up in the equation. Round your answer to one decimal place.</p><p><code class='latex inline'>
\displaystyle
\frac{2}{\sin 50^o} = \frac{8.0}{\sin 60^o}
</code></p>

<p>Charles leaves the marina and sails his boat <code class='latex inline'>10^o</code> west of north for 1.5 h at 18 km/h. He then makes a starboard (right) turn to a heading of <code class='latex inline'>60^o</code> east of north, and sails for 1.2 h at 20 km/h.</p><p>At the end of that time, how far is Charles from his starting point to the nearest kilometre?</p>

<p>For each triangle, determine the value of <code class='latex inline'>\theta</code> to the nearest degree.</p><img src="/qimages/922" />

<p>A bridge across a gorge is 210 m long, as shown in the diagram at the left. The walls of the gorge make angles of 60° and 75° with the bridge. Determine the depth of the gorge to the nearest metre.</p><img src="/qimages/1596" />

<p>Two hot-air balloons are moored to level ground below, each at a different location. An observer at each location determines the angle of elevation to the opposite ballon as shown at the right. The observers are 2.0 km apart.</p><p>(a) What is the distance separating the balloons, to the nearest tenth of a kilometre?</p><img src="/qimages/932" />

<p> Find the area of the shaded figure, correct to two decimals.</p><img src="/qimages/1519" />

<p>Select the most appropriate trigonometric tools among primary trigonometric ratios, the sine law, and the cosine law. Justify your choice. Do not solve.</p><p>In <code class='latex inline'>\triangle XYZ, \angle X = 42^o, y = 25 km</code>, and z = 20km. Determine x.</p>

<p> Use the Law of Sines to find the indicated side x or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/256" />

<p> Sketch the triangle and then solve the triangle using the Law of Sines.</p><p><code class='latex inline'>\angle B = 29^{\circ}</code>, <code class='latex inline'>\angle C = 51^{\circ}</code>, <code class='latex inline'>b = 44</code>.</p>

<p>Two forest fire towers, <code class='latex inline'>A</code> and <code class='latex inline'>B</code>, are 20.3 km apart. From tower <code class='latex inline'>A</code>, the bearing of tower <code class='latex inline'>B</code> is <code class='latex inline'>70^{\circ}</code>. The ranger in each tower observes a fire and radios the bearing of the fire from the tower. The bearing from tower A is <code class='latex inline'>25^{\circ}</code> and from tower <code class='latex inline'>B</code> is <code class='latex inline'>345^{\circ}</code>. How far, to the nearest tenth of a kilometre, is the fire from each tower? </p>

<p>Charles leaves the marina and sails his boat <code class='latex inline'>10^o</code> west of north for 1.5 h at 18 km/h. He then makes a starboard (right) turn to a heading of <code class='latex inline'>60^o</code> east of north, and sails for 1.2 h at 20 km/h.</p><p> What is the course required for Charles to return directly to the marina?</p>

<p> Find the area of the shaded figure, correct to two decimals.</p><img src="/qimages/1521" />

<p>If you want to calculate an unknown side length or angle measure in an acute triangle, what is the minimum information that you must have?</p>

<p> Find the area of the triangle whose sides have the given lengths.</p>
<ul>
<li><code class='latex inline'>a = 7, b = 8, c = 9</code></li>
</ul>

<p> Find the indicated side <code class='latex inline'>x</code> or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/1517" />

<p>Solve for the unknown side length or angle measure. Round your answer to one decimal place.</p><p><code class='latex inline'>
\displaystyle
\frac{k}{\sin 43^o} = \frac{9.5}{\sin 85^o}
</code></p>

<p>In <code class='latex inline'>\angle ABC</code>, two sides and an angle are given. Determine the value of AC to the nearest degree and the length of <code class='latex inline'>b</code> to the nearest tenth
of a centimetre.</p><p><code class='latex inline'>a = 8.6 cm, c = 9.6 cm, \angle A = 47^o</code></p>

<p>Determine each unknown side length to the nearest tenth.</p><img src="/qimages/921" />

<p>In <code class='latex inline'>\triangle PQR, \angle Q = 90^o, r = 6</code>, and <code class='latex inline'>p = 8</code>. Explain two different ways to calculate the measure of <code class='latex inline'>\angle P</code>.</p>

<p> Use the Law of Sines to find the indicated side x or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/252" />

<p>In <code class='latex inline'>\triangle DEF</code>, <code class='latex inline'>DE = 30 cm</code>, <code class='latex inline'>\angle D = 80^{\circ}</code>, and <code class='latex inline'>\angle E = 55^{\circ}</code>.</p><p><strong>(a)</strong> Determine the perimeter of <code class='latex inline'>\triangle DEF</code>, correct to one decimal.</p><p><strong>(b)</strong> Determine the area of <code class='latex inline'>\triangle DEF</code>, correct to one decimal.</p>

<p> Sketch the triangle and then solve the triangle using the Law of Sines.</p><p><code class='latex inline'>\angle A = 50^{\circ}</code>, <code class='latex inline'>\angle B = 68^{\circ}</code>, <code class='latex inline'>c = 230</code>.</p>

<p>An isosceles triangle has two sides that are 10 cm long and two angles that measure 50°. A line segment bisects one of the 50° angles and ends at the opposite side. Determine the length of the line segment.</p>

<p>Matt claims that if <code class='latex inline'>a</code> and <code class='latex inline'>b</code> are adjacent sides in an acute triangle, then a <code class='latex inline'>\sin B = 5 \sin A</code>. Do you agree or disagree? Justify your decision.</p>

<p>Use the Law of Cosines to determine the indicated side x or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/1510" />

<p>Use the Law of Cosines to determine the indicated side x or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/1511" />

<p>A telephone pole is supported by two wires on opposite sides.
At the top of the pole, the wires form an angle of 60°. On the ground, the ends of the wires are 15.0 m apart. One wire makes a 45° angle with the ground. How long are the wires, and how tall is the pole?</p>

<p> To find the distance across a small lake, a surveyor has taken the measurements shown. Find the distance across the lake using this information.</p><img src="/qimages/1523" />

<p>Use the Law of Cosines to determine the indicated side x or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/1513" />

<p>A decorative pottery bowl with a diameter of 30 cm is used as a garden ornament. A rain shower fills it with water to a maximum depth of 7 cm. The bowl is slowly tipped to remove the water. What angle will the rim of the bowl make with the horizontal when the water begins to spill out?</p>

<p>From the bottom of a canyon, Rita stands 47 m directly below an overhead bridge. She estimates that the angle of elevation of the bridge is about 35° at the north end and about 40° at the south end. For each question, round your answer to the nearest metre.</p><p>a) If the bridge were level, how long would it be?</p><p>b) If the bridge were inclined 4° from north to south, how much longer would it be?</p><img src="/qimages/5299" />

<p>Use the sine law to write a ratio that is equivalent to each expression for <code class='latex inline'>\triangle ABC</code>.</p><p><strong>a)</strong> <code class='latex inline'>\displaystyle \frac{a}{\sin a}</code></p><p><strong>b)</strong> <code class='latex inline'>\displaystyle \frac{\sin A}{\sin B}</code></p><p><strong>c)</strong> <code class='latex inline'>\displaystyle \frac{a}{c}</code></p><p><strong>d)</strong> <code class='latex inline'>\displaystyle \frac{a\sin C}{c\sin A}</code></p>

<p>In <code class='latex inline'>\triangle CAT, \angle C = 32^o, \angle T = 81^o</code>, and <code class='latex inline'>c = 24.1 m</code>. Solve the triangle. </p>

<p>There is a water hazard between a golfer's ball and the green. The gofer has two choices. He can hit the ball alongside the water hazard to a point left of the green and play the next shot from there. Or, he can hit directly over the water hazard to the green. The golfer can usually hit an approach shot at least 60 m. Should he attempt the direct shot, or go around the hazard? </p><img src="/qimages/9864" />

<p> Find the area of the triangle whose sides have the given lengths.</p>
<ul>
<li><code class='latex inline'>a = 11, b = 100, c = 101</code></li>
</ul>

<p>Solve the triangle. Round each length to the nearest unit and each angle to the nearest degree.</p><p><code class='latex inline'>\triangle DEF: \angle D = 67^o, \angle F = 42^o, e = 25</code></p>

<p> Find the indicated side <code class='latex inline'>x</code> or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/1516" />

<p>The posts of a hockey goal are 2.0 m apart. A player attempts to score by shooting the puck along the ice from a point 6.5 m from one post and 8.0 m from the other. Within what angle <code class='latex inline'>\theta</code> must the shot be made? Round your answer to the nearest degree.</p><img src="/qimages/930" />

<p>Determine <code class='latex inline'>w</code> to the nearest tenth.</p><img src="/qimages/924" />

<p>Two hot-air balloons are moored to level ground below, each at a different location. An observer at each location determines the angle of elevation to the opposite ballon as shown at the right. The observers are 2.0 km apart.</p><p>Determine the difference in height (above the ground) between the two balloons. Round your answer to the nearest metre.</p><img src="/qimages/932" />

<p>Jim says that the sine law cannot be used to determine the length of side <code class='latex inline'>c</code> in <code class='latex inline'>\triangle ABC</code> at the bottom. Do you agree or disagree? Explain.</p><img src="/qimages/1597" />

<p>Given that O is the centre of the circle with radius of 10, find the area of the <code class='latex inline'>\triangle ACD</code> if <code class='latex inline'>\angle AOD = 120^{\circ}</code> and <code class='latex inline'>\overline{OC} = 22</code>.</p><p> <img src="/qimages/260" /></p>

<p>Given <code class='latex inline'>\Delta ABC</code> below, <code class='latex inline'>BC = 2.0</code> and <code class='latex inline'>D</code> is the midpoint of <code class='latex inline'>BC</code>. Determine <code class='latex inline'>AB</code>, to the nearest tenth, if <code class='latex inline'>\angle ADB = 45^{\circ}</code> and <code class='latex inline'>\angle ACB = 30^{\circ}</code>.</p><img src="/qimages/22167" />

<p>Determine the indicated side lengths and angle measures.</p><img src="/qimages/1588" />

<p> Find the area of the shaded figure, correct to two decimals.</p><img src="/qimages/1522" />

<p>Determine the indicated side lengths and angle measures.</p><img src="/qimages/1589" />

<p>Determine <code class='latex inline'>\theta</code> to the nearest degree.</p><img src="/qimages/925" />

<p>For each triangle, determine the value of <code class='latex inline'>\theta</code> to the nearest degree.</p><img src="/qimages/923" />

<p>For each situation, determine all unknown side lengths to the nearest tenth of a centimetre and/or all unknown interior angles to the nearest degree. If more than one solution is possible, state all possible answers.</p><p>(a) A triangle has exactly one angle measuring <code class='latex inline'>45^o</code> and sides measuring 5.0 cm, 7.4 cm, and 10.0 cm.</p>

<p>Determine each unknown side length to the nearest tenth.</p><img src="/qimages/920" />

<p>For each acute triangle,</p><p>write the ratios that are equivalent by copying the triangle and labelling the sides using lower-case letters.</p><img src="/qimages/1584" />

<p>An architect designed a house that is 12.0 m wide. The rafters that hold up the roof are equal in length and meet at an angle of 70°,
as shown at the left. The rafters extend 0.3 In beyond the supporting wall. How long are the rafters?</p><img src="/qimages/1595" />

<p>Use the Law of Cosines to determine the indicated side x or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/1512" />

<p>The short sides of a parallelogram are both 12.0 cm. The acute angles of the parallelogram are <code class='latex inline'>65^o</code>, and the short diagonal is 15.0 cm.
Determine the length of the long sides of the parallelogram. Round your answer to the nearest tenth of a centimetre.</p>

<p>Use the Law of Cosines to determine the indicated side x or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/1509" />

<p>While golfing, Sean hits a tee shot from <code class='latex inline'>T</code> toward a hole at <code class='latex inline'>H</code>, but the ball veers <code class='latex inline'>23^{\circ}</code> and lands at <code class='latex inline'>B</code>. The scorecard says that <code class='latex inline'>H</code> is 270 m from <code class='latex inline'>T</code>. If Sean walks 160 m to the ball (B), how far, to the nearest metre, is the ball from the hole?</p><img src="/qimages/931" />

<p>Determine the indicated side lengths and angle measures.</p><img src="/qimages/1590" />

<p>Use the Law of Cosines to determine the indicated side x or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/1514" />

<p>Determine the indicated unknown quantity:</p><p> <code class='latex inline'>\triangle DEF, \angle D = 60^o, \angle F = 50^o</code>, and d = 12 cm. Determine <code class='latex inline'>f</code>.</p>

<p>Determine the indicated side lengths and angle measures.</p><img src="/qimages/1590" />

<p> Find the area of the triangle whose sides have the given lengths.</p>
<ul>
<li><code class='latex inline'>a = 9, b = 12, c = 15</code></li>
</ul>

<p>For each situation, determine all unknown side lengths to the nearest tenth of a centimetre and/or all unknown interior angles to the nearest degree. If more than one solution is possible, state all possible answers.</p><p>(b) An isosceles triangle has at least one interior angle of <code class='latex inline'>70^o</code> and at least one side of length 11.5 cm.</p>

<p>Determine the indicated measures to one decimal place.</p><img src="/qimages/1586" />

<p>Calculate each unknown angle to the nearest degree and each unknown In length to the nearest tenth of a centimetre.</p><img src="/qimages/927" />

<p>The interior angles of a triangle are <code class='latex inline'>120^{\circ}</code>, <code class='latex inline'>40^{\circ}</code>, and <code class='latex inline'>20^{\circ}</code>. The longest side is 10 cm longer than the shortest side. Determine the perimeter of the triangle to the nearest centimetre. </p>

<p> Use the Law of Sines to find the indicated side <code class='latex inline'>x</code> or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/255" />

<p> Find the area of the triangle whose sides have the given lengths.</p>
<ul>
<li><code class='latex inline'>a = 1, b = 2, c = 2</code></li>
</ul>

<p>Bill and Nadia live across a ravine from each other. Bill rowed 180 m to the end of the ravine, turned right through an angle of <code class='latex inline'>45^o</code>, and walked another <code class='latex inline'>200 m</code> to Nadia's house. Determine an exact expression for the distance between the two houses.</p>

<p>In <code class='latex inline'>\Delta ABC</code>, <code class='latex inline'>a = 11.5</code>, <code class='latex inline'>b = 8.3</code>, and <code class='latex inline'>c = 6.6</code>. Calculate <code class='latex inline'>\angle A</code> to the nearest degree.</p>

<p>Determine the indicated side lengths and angle measures.</p><img src="/qimages/1593" />

<p> Find the area of the shaded figure, correct to two decimals.</p><img src="/qimages/1520" />

<p>For each acute triangle,</p>
<ul>
<li><em>i)</em> copy the triangle and label the sides using lower-case letters</li>
<li><em>ii)</em> write the ratios that are equivalent</li>
</ul>
<img src="/qimages/1583" />

<p>Select the most appropriate trigonometric tools among primary trigonometric ratios, the sine law, and the cosine law. Justify your choice. Do not solve.</p><p>In <code class='latex inline'>\triangle DEF, \angle D = 60^o, \angle F = 50^o</code>, and d = 12 cm. Determine f.</p>

<p>In <code class='latex inline'>\Delta PQR</code>, <code class='latex inline'>q = 25.1</code>, <code class='latex inline'>r = 71.3</code>, and <code class='latex inline'>\cos P = \frac{1}{4}</code>. Calculate <code class='latex inline'>p</code> to the nearest tenth.</p>

<p> Find the indicated side <code class='latex inline'>x</code> or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/1518" />

<p>Determine the indicated unknown quantity:</p><p>In <code class='latex inline'>\triangle XYZ, \angle X = 42^o, y = 25 km</code>, and <code class='latex inline'>z = 20km</code>. Determine <code class='latex inline'>x</code>.</p>

<p>A Ferris wheel has a radius of 20 m, with 10 cars spaced around the circumference at equal distances. If the cars are numbered in order, how far is it directly from the first car to the fifth car? </p>

<p>Solve for the unknown side length or angle measure. Round your answer to one decimal place.</p><p><code class='latex inline'>
\displaystyle
\frac{12.5}{\sin Y} = \frac{12.5}{\sin 88^o}
</code></p>

<p> In any <code class='latex inline'>\triangle ABC</code>, constants <code class='latex inline'>k</code>, <code class='latex inline'>m</code>, and <code class='latex inline'>p</code> exist so that <code class='latex inline'>k\sin A + m\sin B + p\sin C = 0</code>. Prove that <code class='latex inline'>ka + mb + pc = 0</code>, where <code class='latex inline'>a</code>, <code class='latex inline'>b</code>, <code class='latex inline'>c</code> are the lengths of the sides of the triangle.</p>

<p>Determine the indicated side lengths and angle measures.</p><img src="/qimages/1592" />

<p>The Leaning Tower of Pisa is 55.9 m tall and leans <code class='latex inline'>5.5^{\circ}</code> from the vertical. If its shadow is 90.0 m long, what is the distance from the top of the tower to the top edge of its shadow? Assume that the ground around the tower is level. Round your answer to the nearest metre.</p>

<p>Sean is a naturalist. He is studying the effects of acid rain on fish populations in different lakes. As part of his research, he needs to know the length of Lake Lebarge. Scott makes the measurements shown. How long is Lake Lebarge?</p><img src="/qimages/1594" />

<p>For each of the following, draw possible diagrams that match the given measurements. Then, calculate the length of side <code class='latex inline'>c</code>. If the calculation cannot be made, explain why. </p><p>In <code class='latex inline'>\angle ABC, a = 24 m, b = 21 m</code>, and <code class='latex inline'>\angle A = 75^o</code>. </p><p>Select the right choice:</p>

<p>Write three equivalent ratios using the sides and angles in the triangle at the right.</p><img src="/qimages/1585" />

<p>Solve for the unknown side length or angle measure. Round your answer to one decimal place.</p><p><code class='latex inline'>
\displaystyle
\frac{6.3}{\sin M} = \frac{10.0}{\sin 72^o}
</code></p>

<p>You receive a scientific calculator at checkpoint num 3. Determine the direction and distance to checkpoint num 4 from the information below. Draw the leg on your map. Include all angles and distances.</p><p><strong>Direction:</strong> North of West</p>
<ul>
<li>Use <code class='latex inline'>\angle A</code> from <code class='latex inline'>\triangle ABC</code>. In <code class='latex inline'>\triangle ABC</code>, <code class='latex inline'>\angle B = 85^o</code>, <code class='latex inline'>a = 41 m</code>, and <code class='latex inline'>c = 32 m</code>. Round to the nearest degree, if necessary. </li>
</ul>
<p><strong>Distance</strong> The measure of <code class='latex inline'>b</code>, in <code class='latex inline'>\triangle ABC</code>, to the nearest metre.</p>

<p>Draw a labelled diagram for the triangle. Then calculate the required side length or angle measure.</p>
<ul>
<li>In <code class='latex inline'>\triangle SUN</code>, <code class='latex inline'>n = 58 cm</code>, <code class='latex inline'>\angle N = 38^o</code>, and <code class='latex inline'>\angle U = 72^o</code>. Determine the length of side <code class='latex inline'>u</code>.</li>
</ul>

<p>In <code class='latex inline'>\triangle ABC, \angle A = 58^o, \angle C = 74^o</code>, and <code class='latex inline'>b = 6</code>. Calculate the area of <code class='latex inline'>\triangle ABC</code> to one decimal place.</p>

<p>Draw a labelled diagram for the triangle. Then calculate the required side length or angle measure.</p>
<ul>
<li>In <code class='latex inline'>\triangle PQR</code>, <code class='latex inline'>\angle R = 73^o</code>, <code class='latex inline'>\angle Q = 32^o</code>, and <code class='latex inline'>r = 23 cm</code>. Determine the length of side <code class='latex inline'>q</code>.</li>
</ul>

<p>Determine the indicated measures to one decimal place.</p><img src="/qimages/1587" />

<p>Draw a labelled diagram for the triangle. Then calculate the required side length or angle measure.</p>
<ul>
<li>In <code class='latex inline'>\triangle TAM</code>, <code class='latex inline'>t = 8 cm</code>, <code class='latex inline'>m = 6 cm</code>, and <code class='latex inline'>\angle T = 65^o</code>. Determine the measure of <code class='latex inline'>\angle M</code>.</li>
</ul>

<p> Use the Law of Sines to find the indicated side x or angle <code class='latex inline'>\theta</code>.</p><img src="/qimages/253" />

<p>Calculate each unknown angle to the nearest degree and each unknown In length to the nearest tenth of a centimetre.</p><img src="/qimages/926" />