Find angles from sine, cosine and tangent measures
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!
$\sin\theta=\frac{\text{Opposite }}{\text{Hypotenuse }}$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 $\sin^{-1}$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.