4.01 Review of trigonometry for right-angled triangles

For each of the following right-angled triangles, evaluate:

a

b

Consider the following triangle:

a

Find the value of \sin \left(90 \degree - \theta\right).

b

Find the value of \cos \theta.

c

What do you notice?

Consider the following triangle:

a

Find the value of \cos \left(90 \degree - \theta\right).

b

Find the value of \sin \theta.

c

What do you notice?

For each of the following right-angled triangles, find the value of:

a

b

For each of the following right-angled triangles, find the value of:

a

b

Find the value of the pronumeral in the following triangles, correct to two decimal places:

a

b

c

d

e

f

g

h

i

j

Given the value of \tan \theta, find the value of b in the following diagrams. Round your answers to one decimal place if necessary.

a

\tan \theta = 0.4

b

\tan \theta = \dfrac{2}{3}

Find \theta, to one decimal place, given 0 \degree \leq \theta \leq 90 \degree:

Find \theta, to the nearest degree, given 0 \degree \leq \theta \leq 90 \degree:

For each of the following right-angled triangles, find the value of x to the nearest degree:

a

b

c

State the exact value of the following:

Given that \sin \theta = \dfrac{1}{2} in a right triangle:

Find the exact value of the pronumeral(s) in the following triangles:

a

b

c

d

e

f

g

h

i

j

Use exact values to solve for \theta in the following triangles:

a

b

For each of the following isosceles triangle, calculate the size of angle A to one decimal place:

a

b

A 13.7 \text{ m} long string of lights joins the top of a tree to a point on the ground. If the tree is 3.7 \text{ m} tall, find \theta, the angle the string of lights would make with the tree. Round your answer to two decimal places.

Georgia is riding her pushbike up a hill that has an incline of 7 \degree. If she rides her bike 689 \text{ m} up the hill, find the horizontal distance from her starting point, to the nearest metre.

From the top of a rocky ledge 188 \text{ m} high, the angle of depression to a boat is 13 \degree. If the boat is d \text{ m} from the foot of the cliff find d correct to 2 decimal places.

Buzz is standing 49 \text{ m} from a building and measures the angle of elevation of the top of the building to be 23 \degree.

Lisa is on a ship and observes a lighthouse on a cliff in the distance. The base of the cliff is 906 \text{ m} away from the ship, and the angle of elevation of the top of the lighthouse from Lisa is 16 \degree.

Consider the following diagram:

a

Find the following to two decimal places:

i

y

ii

w

b

Hence, find x. Round your answer to one decimal place.

A fighter jet, flying at an altitude of 2000 \text{ m} is approaching an airport. The pilot measures the angle of depression to the airport to be 13 \degree. One minute later, the pilot measures the angle of depression again and finds it to be 16 \degree.

a

Find the distance AC, to the nearest metre.

b

Find the distance BC, to the nearest metre.

c

Hence, find the distance covered by the jet in that one minute, to the nearest metre.

A jet takes off and leaves the runway at an angle of 34 \degree. It continues to fly in this direction for 7 \text{ min} at a speed of 630 \text{ km/h} before levelling out.

a

Find the distance covered by the jet just before levelling out in metres.

b

If the height of the jet just before levelling off is h \text{ m}, calculate h. Round your answer to the nearest metre.

A safety fence is constructed to protect tourists from the danger of an eroding castle toppling down. The surveyor takes an angle measurement to the top of the tower of 10 \degree. She then walks 29 \text{ m} towards the tower and takes another reading of 22 \degree.

Find the value of h to the nearest metre.