label the sides as O, A or H with respect to the position of the angle you want to find
identify the appropriate trigonometric ratio that applies [either sine (sin), cosine (cos) or tangent (tan)]
using algebra, solve the equation for the angle, (write the rule, fill in what you know, then solve using inverse operations)
reflect and check (do a quick check on your calculator to confirm your answer is correct)

Examples

Question 1

Find the angle indicated in this diagram.

1. label the sides as O, A or H with respect to the position of the angle

2. Identify the appropriate ratio that uses O and H. For this question it will be sine (sin)

3. Using algebra, solve the equation for the angle

(write the rule) $\sin\theta=\frac{O}{H}$sinθ=OH

(fill in what you know) $\sin\theta=\frac{5}{8}$sinθ=58

(solve using inverse operations) $\theta=\sin^{-1}\left(\frac{5}{8}\right)$θ=sin−1(58) use a calculator for this bit!

$\theta=38.68$θ=38.68°

Using the inverse operation for sin/cos/tan

We looked already at calculating angles from a value, here is a reminder.

Find $\theta$θ if $\sin\theta=0.65$sinθ=0.65 answer to $2$2 decimal places

This question is asking us what the angle is if the ratio of the opposite and hypotenuse is $0.65$0.65. To answer this question you use the inverse sin button on your scientific calculator. Often it looks a bit like this $\sin^{-1}$sin−1

$\sin\theta=0.65$sinθ=0.65

$\sin^{-1}$sin−1 $0.65=40.54$0.65=40.54°

Question 2

Find the value of the angle indicated.

We have the opposite and adjacent sides here, so the ratio I will use is tangent (tan).

$\tan\theta=\frac{O}{A}$tanθ=OA write the rule

$\tan\theta=\frac{14.77}{12.24}$tanθ=14.7712.24 fill in what we know

$\theta=\tan^{-1}\left(\frac{14.77}{12.24}\right)$θ=tan−1(14.7712.24) use inverse operations to rearrange, and then use a calculator

$\theta=50.35$θ=50.35°

Question 3

Find the value of $x$x to the nearest degree.

A right-angled triangle with vertices labeled $A$ , $B$ and $C$ . Vertex $A$ is at the top, $B$ at the bottom right, and $C$ at the bottom left. A small square at vertex $A$ indicates that it is a right angle. Side $interval(BC)$ , which is the side opposite vertex $A$ , is the hypotenuse and is marked with a length of $25$ . The angle located at vertex B is labelled $x$ . Side $interval(AB)$ , descending from the right angle at vertex $A$ to vertex $B$ , is marked with a length of $7$ , and is adjacent to the angle $x$ . Side $interval(AC)$ is opposite the angle $x$ .

Question 4

Consider the given figure.

Find the unknown angle $x$x, correct to two decimal places.
Find $y$y, correct to two decimal places.
Find $z$z correct to two decimal places.

Question 5

The person in the picture sights a pigeon above him. If the angle the person is looking at is $\theta$θ, find $\theta$θ in degrees.

Round your answer to two decimal places.

Outcomes

10P.MT2.02

Determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean Theorem

Find angles in right-angled triangles