# Find angles from sine, cosine and tangent measures

Lesson

As with every topic in mathematics, there is a conceptual side (what you need to know and understand) and a practical side (what you need to do and answer). To calculate values involving trigonometric expressions, it will often be easiest to use a scientific calculator.

Don't forget the three trigonometric ratios!

Trigonometric ratios

$\sin\theta=\frac{\text{Opposite }}{\text{Hypotenuse }}$sinθ=Opposite Hypotenuse

$\cos\theta=\frac{\text{Adjacent }}{\text{Hypotenuse }}$cosθ=Adjacent Hypotenuse

$\tan\theta=\frac{\text{Opposite }}{\text{Adjacent }}$tanθ=Opposite Adjacent

#### Worked example

##### Question 1

If $\sin\theta=0.65$sinθ=0.65, find $\theta$θ to the nearest degree.

Think: This question is asking us what the angle ($\theta$θ) is, if the ratio of the opposite side and hypotenuse is $0.65$0.65. To answer this question, we can use the inverse sine button on a calculator. It will probably look like $\sin^{-1}$sin1, and may involving pressing 'shift' or '2nd F'.

Do:

 $\sin\theta$sinθ $=$= $0.65$0.65 $\theta$θ $=$= $\sin^{-1}\left(0.65\right)$sin−1(0.65) (Take the inverse sine) $\theta$θ $=$= $40.54160187$40.54160187$\ldots$… (Evaluate with a calculator) $\theta$θ $=$= $41$41 (Round to the nearest degree)

#### Practice questions

##### QUESTION 1

If $\cos\theta=0.146$cosθ=0.146, find $\theta$θ, writing your answer to the nearest degree.

##### QUESTION 2

If $\sin\theta=1$sinθ=1, find $\theta$θ.

##### QUESTION 3

If $\tan\theta=1.732$tanθ=1.732, find $\theta$θ, writing your answer to the nearest degree.

### Outcomes

#### 10D.T2.02

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