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Solutions
22 Videos

Use a unit circle to determine exact values for the primary trigonometric ratios for 210°.

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Q1

A ship is tied to a dock with a rope of length 10 m. At low tide, the rope is stretched tight, forming an angle of 45° with the horizontal. At high tide, the stretched rope makes an angle of 30° with the horizontal. How much closer to the dock, horizontally, is the ship at low tide than at high tide? Determine an exact expression. Then, use a calculator to determine an approximate answer, correct to the nearest tenth of a metre.

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Q2

The coordinates of a point on the terminal arm of an angle `\theta`

are shown. Determine the exact primary trigonometric ratios for `\theta`

.

`A(-5, 12)`

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Q3a

The coordinates of a point on the terminal arm of an angle `\theta`

are shown. Determine the exact primary trigonometric ratios for `\theta`

.

`(3, -4)`

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Q3b

The coordinates of a point on the terminal arm of an angle `\theta`

are shown. Determine the exact primary trigonometric ratios for `\theta`

.

`(6, -8)`

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Q3c

One of the primary trigonometric ratios for an angle is given, as well as the quadrant that the terminal arm lies in. Determine the other two primary trigonometric ratios.

`\sin A = \frac{4}{5}`

, first quadrant

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Q4a

One of the primary trigonometric ratios for an angle is given, as well as the quadrant that the terminal arm lies in. Determine the other two primary trigonometric ratios.

`\cos B = \frac{8}{17}`

, fourth quadrant

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Q4b

One of the primary trigonometric ratios for an angle is given, as well as the quadrant that the terminal arm lies in. Determine the other two primary trigonometric ratios.

`\tan C = -\frac{12}{5}`

, second quadrant

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Q4c

`\sin D = -\frac{4}{7}`

, third quadrant

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Q4d

Find `\theta`

accurate to nearest degree.

Solve the equation `\sin \theta = -0.25`

for `0^o \leq \theta \leq 360^o`

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Q5a

Find `\theta`

accurate to nearest degree.

Solve the equation `\cos \theta = \frac{4}{5}`

for `0^o \leq \theta \leq 360^o`

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Q5b

Find `\theta`

accurate to nearest degree.

Solve the equation `\tan \theta = \frac{5}{8}`

for `0^o \leq \theta \leq 360^o`

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Q5c

The point lies on the terminal arm of an angle. Determine the six trigonometric ratios for the angle, rounded to four decimal places.

```
\displaystyle
(7, -24)
```

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Q6d

Determine two angles between 0° and 360° that have a secant of -4. Round your answers to the nearest degree.

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Q7

An angle between 0° and 360° has a cosecant of `-1`

.

a) Is this enough information to determine a unique solution? If yes, explain why. If no, what other information is required?

b) Determine the angle or angles.

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Q8

Marko rides 10 km north of his home on his mountain bike. He reaches an abandoned railroad, turns through an angle of 120° onto the railroad, and then rides another 20 km.

a) Draw a diagram to model this situation, labelling all distances and angles.

b) Select the most appropriate trigonometric tool to determine Marko’s distance from home. Justify your selection.

c) Determine an exact value for Marko’s distance from home. Then, use a calculator to find the approximate distance.

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Q9

In `\triangle ABC`

. `\angle A`

= 32°, `a = 15`

m, and `b = 18`

m.

a) Use drawing tools or geometry software to illustrate why the ambiguous case applies to this situation

b) Sketch diagrams to represent the two possible triangles that match these measurements.

c) Solve for side C in both triangles, to the nearest metre.

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Q10

An asymmetric pyramid has a base in the shape of a kite, with the longer sides of the base measuring 150 m and the shorter sides measuring 120 m. The angle between the two longer sides measures 70°. The angle of elevation of the top of the pyramid, as seen from the vertex between the longer and shorter sides, is 75°. Determine the height of the pyramid to the nearest tenth of a kilometre.

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Q11

Cate is sailing her boat off the coast, which runs straight north and south. Her GPS confirms that she is 8 km from Haytown and 10 km from Beeville, two towns on the coast. The towns are separated by an angle of `80^o`

, as seen from the boat. A helicopter is hovering at an altitude of 1000 m halfway between Haytown and Beeville.

a) Determine the distance between Haytown and Beeville, to the nearest tenth of a kilometre.

b) Determine the angle of elevation of the helicopter, as seen from the sailboat, to the nearest tenth of a degree.

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Q12

Prove that `\frac{\cot \theta}{\csc \theta } = \cos \theta`

.

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Q13

Prove that `\sin^4\theta - \cos^4\theta = \sin^2\theta - \cos^2\theta`

.

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Q14

Prove that ```
\displaystyle
\cot\theta = \cos\theta \sin \theta + \cos^3\theta\csc\theta
```

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Q15