Angles

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inverses

We have two types of trigonometric functions.

We have functions that map angles to ratios:

\[ \sin (\theta ), \cos (\theta ) : \text { Angles } \to \text { Ratios } \]

And, we have their inverses:

\[ \sin ^{-1}(t), \cos ^{-1}(t) : \text { Ratios } \to \text { Angles } \]

Your calculator has buttons for all of these.

Note: Your calculator has two modes for angle measurements: degrees and radians. Remember to switch between these modes when calculating.

Calculator

Approximate the following.

$\cos (20^{\circ }) \approx \answer [tolerance=0.001]{0.9396926208}$
$\sin (65^{\circ }) \approx \answer [tolerance=0.001]{0.906307787}$
$\cos \left ( \frac {\pi }{3} \right ) \approx \answer [tolerance=0.001]{0.5}$
$\sin \left ( \frac {\pi }{4} \right ) \approx \answer [tolerance=0.001]{0.7071067812}$

Your calculator also has buttons for the inverse functions.

You can also supply the calculator with the ratio and the calculator returns the angle.

Calculator

Approximate the following.

$\cos ^{-1}(0.9396926208) \approx \answer [tolerance=0.001]{20}^{\circ }$
$\sin ^{-1}(0.906307787) \approx \answer [tolerance=0.001]{65}^{\circ }$
$\cos ^{-1}(0.5) \approx \answer [tolerance=0.001]{1.047197551}$
$\sin ^{-1}(0.7071067812) \approx \answer [tolerance=0.001]{0.7853981634}$

Other Quadrants

Remember that the signs of sine and cosine changes as we move through the four quadrants. We introduced a reference angle and a refrence triangle to help with lengths and then we also had to think of the sign.

The calculator also knows that the signs change.

The calculator will give the correct values of sine and cosine along with their proper sign. However, going backwards is a different story.

If you give the calculator a value of sine and ask it for the angle, then there are an infinite number of angles that have that same value of sine. Just keep spinning around the circle.

$\blacktriangleright $ Given $\sin (\theta ) = \frac {1}{2}$, then any of the following are possible.

$\theta = -\frac {7\pi }{6}$
$\theta = \frac {\pi }{6}$
$\theta = \frac {5\pi }{6}$
$\theta = \frac {13\pi }{6}$
$\theta = \frac {17\pi }{6}$
an infinite number of other angles

Therefore, the calculator just focuses on two quadrants, one for each sign. The calculate gives angles in the fourth and first quadrants, $\left ( -\frac {\pi }{2}, \frac {\pi }{2} \right )$, when sine is negative or positive. Then it is up to you to figure out which angle you really want from there.

Backwards

$150^{\circ }$ is an angle in the second quadrant. $\sin (150^{\circ }) = \frac {1}{2} = 0.5$.

Now, ask your calculator to give you an angle whose sine is $0.5$.

$\sin ^{-1}(0.5) = \answer {30}^{\circ }$

It doesn’t give $150^{\circ }$. It gives $30^{\circ }$.

We’ll study this more later in the course. The short story is we have trigonometric functions and their inverse functions.

$\sin ^{-1}(x)$ will give angles in the fourth and first quadrants.
$\cos ^{-1}(x)$ will give angles in the first and second quadrants.
$\tan ^{-1}(x)$ will give angles in the fourth and first quadrants.

The above is certainly proper inverse function notation, but it confuses everyone, because it is also proper notation for a reciprocal. Therefore, we prefer other names:

$\arcsin (x)$ will give angles in the fourth and first quadrants.
$\arccos (x)$ will give angles in the first and second quadrants.
$\arctan (x)$ will give angles in the fourth and first quadrants.

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more examples can be found by following this link
More Examples of Right Triangles

2025-05-18 00:59:25