Trigonometric functions

Trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle.

If we have a right triangle with one angle \(\theta\)

Right Triangle

Then the following hold true

$$sine\: \theta =\frac{a}{c}$$

$$cosine \:\theta =\frac{b}{c}$$

$$tangent \:\theta =\frac{a}{b}$$

$$cosecant \:\theta =\frac{c}{a}$$

$$secant \:\theta =\frac{c}{b}$$

$$cotangent \:\theta =\frac{b}{a}$$

Example

If we are given one angle in a right triangle \( \theta = 45^{\circ}\) and we know that the measure of the hypotenuse is 2, what are the measures of the two other legs?

Using the formulas above we get that:

$$sine\; 45^{\circ}=\frac{a}{2}$$

$$cosine\; 45^{\circ}=\frac{b}{2}$$

We use our calculators to determine sine \(45^{\circ}\) and cosine \(45^{\circ}\) :

$$0.707=\frac{a}{2}$$

$$0.707=\frac{b}{2}$$

Now we can see that the measures of \(a\) and \(b\) are equal and that the length is

$$2 \cdot 0.707 = 1.414$$

Video lesson

If we are given one angle in a right triangle \(\theta = 20^{\circ}\) and we know that the measure of the hypotenuse is 3, what are the measures of the two other legs?