4.7 Chapter Review
Chapter
Chapter 4
Section
4.7
Solutions 45 Videos

For each angle, use exact values to show that \sin^2 x + \cos^2 x = 1.

x = 60^{\circ}

Q1a

For each angle, use exact values to show that sin^2 x + cos^2 x = 1.

x = 150^{\circ}

Q1b

For each angle, use exact values to show that sin^2 x + cos^2 x = 1.

x = 225^{\circ}

Q1c

Determine the exact value of each expression.

sin 45^{\circ} \times tan 30^{\circ} \times cos 120^{\circ}

Q2a

Determine the exact value of each expression.

sin 240^{\circ} + tan 225^{\circ} - cos 330^{\circ}

Q2b

Determine the exact value of each expression.

sin 270^{\circ} \times tan 300^{\circ} - cos 180^{\circ}

Q2c

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}

Q3a

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}

Q3b

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}

Q3c

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.

Q4

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)

Q5a

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)

Q5b

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

Q6a

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

Q6b

Solve each equation for 0^{\circ} \leq \theta \leq 360^{\circ}.

sin \theta = \dfrac{5}{8}

Q7a

Solve each equation for 0^{\circ} \leq \theta \leq 360^{\circ}.

cos \theta = -0.35

Q7b

Solve each equation for 0^{\circ} \leq \theta \leq 360^{\circ}.

tan \theta = \dfrac{4}{9}

Q7c

Determine exact expressions for the six trigonometric ratios for 120^{\circ}.

Q8

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)

Q9a

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)

Q9b

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)

Q9c

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)

Q9d

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.

Q10a

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.

Q10b

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.

Q10c

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.

Q10d

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.

Q10e

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.

Q10f

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?

Q11

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?

Q12a

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.

Q13

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.

Q13

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?

Q14

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.

Q15

State an equivalent expression for each.

cos^2 \theta

Q16a

State an equivalent expression for each.

csc^2 \theta

Q16b

State an equivalent expression for each.

cot^2 \theta

Q16c

State an equivalent expression for each.

sec^2 \theta - 1

Q16d

State an equivalent expression for each.

sec^2 \theta - tan^2 \theta

Q16e

State an equivalent expression for each.

\dfrac{1}{sec^2 \theta}

Q16f

Prove each identity.

sec^2 \theta + csc^2 \theta = (tan \theta + cot \theta)^2

Q17a

Prove each identity.

\dfrac{sin \theta + cos \theta cot \theta}{cot \theta} = sec \theta

Q17b

Prove each identity.

sin^2 \theta = cos \theta (sec \theta - cos \theta)^2

Q17c

Prove each identity.

\dfrac{1}{1 + cos \theta} = csc^2 \theta - \dfrac{cot \theta}{sin \theta}

\dfrac{1 + csc \theta}{cot \theta} - sec \theta = tan \theta