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Solutions
45 Videos

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

.

`x = 60^{\circ}`

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Q1a

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

.

`x = 150^{\circ}`

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Q1b

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

.

`x = 225^{\circ}`

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Q1c

Determine the exact value of each expression.

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

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Q2a

Determine the exact value of each expression.

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

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Q2b

Determine the exact value of each expression.

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

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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}`

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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}`

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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}`

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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.

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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)

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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)

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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

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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

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Q6b

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

.

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

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Q7a

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

.

`cos \theta = -0.35`

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Q7b

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

.

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

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Q7c

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

.

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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)

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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)

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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)

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Q9c

W(1, 3)

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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.

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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.

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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.

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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.

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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.

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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.

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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?

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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?

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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.

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Q13

`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.

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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?

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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.

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Q15

State an equivalent expression for each.

`cos^2 \theta`

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Q16a

State an equivalent expression for each.

`csc^2 \theta`

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Q16b

State an equivalent expression for each.

`cot^2 \theta`

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Q16c

State an equivalent expression for each.

`sec^2 \theta - 1`

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Q16d

State an equivalent expression for each.

`sec^2 \theta - tan^2 \theta`

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Q16e

State an equivalent expression for each.

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

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Q16f

Prove each identity.

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

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Q17a

Prove each identity.

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

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Q17b

Prove each identity.

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

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Q17c

Prove each identity.

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

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Q17d

Prove each identity.

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

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Q17e