New Zealand
Level 6 - NCEA Level 1

# Finding Unknown Angles

Lesson

We have already seen what the trigonometric ratios are:

We know wow to calculate with them, and how to find the length of unknown sides of right-angled triangles with them.

We can also use the trigonometric ratios to find the size of unknown angles.  To do this we need any 2 of the side lengths.

Find the Size of an Unknown Angle
• 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=sin1(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 $\sin^{-1}$sin1

$\sin\theta=0.65$sinθ=0.65

$\sin^{-1}$sin1 $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=tan1(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.

##### Question 4

Consider the given figure.

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

2. Find $y$y, correct to two decimal places.

3. 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.

1. Round your answer to two decimal places.

### Outcomes

#### GM6-6

Use trigonometric ratios and Pythagoras’ theorem in two and three dimensions

#### 91031

Apply geometric reasoning in solving problems

#### 91032

Apply right-angled triangles in solving measurement problems