Evaluate a trigonometric ratio, and find an angle from its value

To find an angle in degrees from a trigonometric value, which setting and key should be chosen first?

Trigonometric technology in degree mode

A plain trigonometric key evaluates a ratio from an angle. The matching inverse key recovers an angle from a ratio value, and the device must use degree mode. These examples recover acute angles, between 0° and 90°

1. 1Set degree mode A degree symbol in the question requires the device to interpret the angle in degrees \(\displaystyle \sin 37^{\circ}\approx0.6018\)
2. 2Match the key to the task Use a plain key for a ratio value and the matching inverse key for an angle \(\displaystyle \theta=\cos^{-1}\left(0.5\right)=60^{\circ}\)
3. 3Treat the inverse as a function The superscript -1 in a trigonometric inverse is not a reciprocal power \(\displaystyle \sin^{-1}\left(0.5\right)\ne\frac{1}{\sin\left(0.5\right)}\)

Evaluate the sine value in degree mode to 4 decimal places

Use the displayed sine value to find the acute angle to the nearest degree

Recover an acute angle from a stated ratio

For an acute angle, choose the inverse key matching the stated ratio. Enter the whole given value, keep degree mode and round only the resulting angle

1. 1Sine Inverse sine recovers the acute angle with this sine value \(\displaystyle \sin\theta=0.5\;\Longrightarrow\;\theta=\sin^{-1}(0.5)=30^{\circ}\)
2. 2Cosine The same ratio value gives a different angle when the named function is cosine \(\displaystyle \cos\theta=0.5\;\Longrightarrow\;\theta=\cos^{-1}(0.5)=60^{\circ}\)
3. 3Tangent A tangent ratio may be 1 or greater; it is not restricted to values below 1 \(\displaystyle \tan\theta=1\;\Longrightarrow\;\theta=45^{\circ}\)

An acute angle has cosine 0.6. Find the angle in degrees to 1 decimal place

An acute angle has tangent 2. Which calculation finds its angle in degrees?

An acute angle has sine 0.7. Find the angle in degrees to 1 decimal place

Which line correctly evaluates the degree angle?

Angles beyond 90 degrees

For the senior supplementary-angle questions, consider the full stated interval

1. 1Check angles beyond 90 degrees For an acute angle, sine is positive at its supplement, while cosine is negative at the supplementary obtuse angle \(\displaystyle \sin\left(180^{\circ}-\theta\right)=\sin\theta,\quad\cos\left(180^{\circ}-\theta\right)=-\cos\theta\)