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.

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 $\sin^{-1}$sin−1, 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 $\tan\theta=1.711$tanθ=1.711, 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