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

10.T.IT.1

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° and 90°. Values (with proofs) of the trigonometric ratios of 30°, 45° and 60°. Relationships between the ratios.

10.T.IT.2

Trigonometric Identities: Proof and applications of the identity sin^2 A + cos^2 A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.

Find angles in right-angled triangles