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 θ

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 θ = 45° 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° and cosine 45°:

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

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

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 θ = 20° and we know that the measure of the hypotenuse is 3, what are the measures of the two other legs?