For each angle, use exact values to show that sin^2 x + cos^2 x = 1.
x = 60^{\circ}
For each angle, use exact values to show that sin^2 x + cos^2 x = 1.
x = 150^{\circ}
For each angle, use exact values to show that sin^2 x + cos^2 x = 1.
x = 225^{\circ}
Determine the exact value of each expression.
sin 45^{\circ} \times tan 30^{\circ} \times cos 120^{\circ}
Determine the exact value of each expression.
sin 240^{\circ} + tan 225^{\circ} - cos 330^{\circ}
Determine the exact value of each expression.
sin 270^{\circ} \times tan 300^{\circ} - cos 180^{\circ}
A guy wire is fastened 6 m from the base of a flagpole and makes an angle \theta with the ground. For each given angle \theta, determine
i) the length of the wire
ii) how far up the pole the wire is fastened
45^{\circ}
A guy wire is fastened 6 m from the base of a flagpole and makes an angle \theta with the ground. For each given angle \theta, determine
i) the length of the wire
ii) how far up the pole the wire is fastened
30^{\circ}
A guy wire is fastened 6 m from the base of a flagpole and makes an angle \theta with the ground. For each given angle \theta, determine
i) the length of the wire
ii) how far up the pole the wire is fastened
60^{\circ}
A tiling company develops a floor tile in the shape of a regular hexagon that has an area of 30 cm^2. Determine the exact length of the six equal sides of the tile.
The coordinates of a point on the terminal arm of an angle \theta are shown. Determine the exact trigonometric ratios for \theta.
E(-5, 12)
The coordinates of a point on the terminal arm of an angle \theta are shown. Determine the exact trigonometric ratios for \theta.
F(8, -6)
One of the primary trigonometric ratios for an angle is given, as well as the quadrant in which the terminal arm lies. Find the other two primary trigonometric ratios.
sin G = -\dfrac{5}{11}, third quadrant
One of the primary trigonometric ratios for an angle is given, as well as the quadrant in which the terminal arm lies. Find the other two primary trigonometric ratios.
cos E = \dfrac{4}{7}, first quadrant
Solve each equation for 0^{\circ} \leq \theta \leq 360^{\circ}.
sin \theta = \dfrac{5}{8}
Solve each equation for 0^{\circ} \leq \theta \leq 360^{\circ}.
cos \theta = -0.35
Solve each equation for 0^{\circ} \leq \theta \leq 360^{\circ}.
tan \theta = \dfrac{4}{9}
Determine exact expressions for the six trigonometric ratios for 120^{\circ}.
Each point lies on the terminal arm of an angle in standard position. Determine exact expressions for the six trigonometric ratios for each angle.
T(24, -7)
Each point lies on the terminal arm of an angle in standard position. Determine exact expressions for the six trigonometric ratios for each angle.
U(-5, -3)
Each point lies on the terminal arm of an angle in standard position. Determine exact expressions for the six trigonometric ratios for each angle.
V(4, -9)
Each point lies on the terminal arm of an angle in standard position. Determine exact expressions for the six trigonometric ratios for each angle.
W(1, 3)
For each triangle, determine the number of solutions then solve the triangle if possible.
In \triangle ABC, \angle A = 71^{\circ}, a = 12.2 m, and b = 11.4 m.
For each triangle, determine the number of solutions then solve the triangle if possible.
In \triangle ABC, \angle A = 55^{\circ}, a = 7.1 cm, and b = 9.6 cm.
For each triangle, determine the number of solutions then solve the triangle if possible.
In \triangle ABC, \angle A = 44^{\circ}, a = 9.3 mm, and b = 12.3 mm.
For each triangle, determine the number of solutions then solve the triangle if possible.
In \triangle DEF, \angle D = 42^{\circ}, d = 8.5 km, and f = 7.3 km.
For each triangle, determine the number of solutions then solve the triangle if possible.
In \triangle DEF, \angle E = 38^{\circ}, d = 16.6 mm, and e = 13.4 mm.
For each triangle, determine the number of solutions then solve the triangle if possible.
In \triangle DEF, \angle D = 47^{\circ}, d = 8.1 m, and f = 12.2 m.
A marathon swimmer starts at Island A, swims 9.2 km to Island B, then 8.6 km to Island C. The angle formed by a line from Island B to Island A and a line from Island C to Island A is 52^{\circ}. How far does the swimmer have to swim to return directly to Island A?
A solar-heated house is 10 m wide. The south side of the roof, containing the solar collectors, rises for 8 m at an angle of elevation of 60^{\circ}.
a) Determine the length of the north side of the roof.
b) At what angle of elevation does the north side of the roof rise?
The pilot of a hot-air balloon, flying above a bridge, measures the angles of depression to each end of the bridge to be 54^{\circ} and 71^{\circ}. The direct distance from the balloon to the nearer end of the bridge is 270 m. Determine the length of the bridge.
The pilot of a hot-air balloon, flying above a bridge, measures the angles of depression to each end of the bridge to be 54^{\circ} and 71^{\circ}. The direct distance from the balloon to the nearer end of the bridge is 270 m. Determine the length of the bridge.
Two roads intersect at an angle of 34^{\circ}. Car A leaves the intersection on one of the roads and travels at 80 km/h. At the same time, Car B leaves the intersection on the other road and travels at 100 km/h. Two hours later a reporter in a traffic helicopter, which is above and between the two cars, notes that the angle of depression to the slower car is 20^{\circ} and that the helicopter is 1 km from that vehicle. How far is the faster car from the helicopter?
The port of Math Harbour is located 200 km from Trig Town Inlet in a direction 50^{\circ} east of north. A yacht leaves Trig Town Inlet at 8:00 am. and sails in a direction 15^{\circ} west of north at a speed of 15 km/h. At the same time, a fishing boat leaves Math Harbour on a course 80^{\circ} west of south at a speed of 20 km/h. Determine the distance, to the nearest kilometre, between the yacht and the fishing boat at 1:00 pm.
State an equivalent expression for each.
cos^2 \theta
State an equivalent expression for each.
csc^2 \theta
State an equivalent expression for each.
cot^2 \theta
State an equivalent expression for each.
sec^2 \theta - 1
State an equivalent expression for each.
sec^2 \theta - tan^2 \theta
State an equivalent expression for each.
\dfrac{1}{sec^2 \theta}
Prove each identity.
sec^2 \theta + csc^2 \theta = (tan \theta + cot \theta)^2
Prove each identity.
\dfrac{sin \theta + cos \theta cot \theta}{cot \theta} = sec \theta
Prove each identity.
sin^2 \theta = cos \theta (sec \theta - cos \theta)^2
Prove each identity.
\dfrac{1}{1 + cos \theta} = csc^2 \theta - \dfrac{cot \theta}{sin \theta}
Prove each identity.
\dfrac{1 + csc \theta}{cot \theta} - sec \theta = tan \theta