Mathematics Study Guide for the ACT

Trigonometry

Trigonometry deals with the relationships between the sides and angles of triangles. It involves the study of trigonometric ratios, functions, and identities.

Unit Circle Trigonometry

Unit circle trigonometry is based on the unit circle, a circle with a radius of \(1\). This circle is the reference for defining trigonometric functions and their values for any angle. It is a circle centered at the origin of a coordinate grid. It is divided into four quadrants, and the angles are measured in degrees or radians. The coordinates of the points on the unit circle correspond to the values of trigonometric functions for the angles formed by the terminal side of the angle and the positive \(x\)-axis.

Trigonometric Ratios

The basic three trig ratios are sine, cosine, and tangent. These ratios are defined based on the angles of a right triangle. In a right triangle, the hypotenuse is the side opposite to the right angle. The other two sides are the legs. Refer to the diagram below:

34 Triangle Sides.png

Imagine standing on the angle of interest, \(x\), looking toward the interior of the triangle. The leg directly in front of you is the opposite leg. The leg to your side is the adjacent leg.

Now, we can define the three trig ratios and show their graphs as follows:

\[\sin{x} = \frac{\text{opposite}}{\text{hypotenuse}}\]

35 Sine Graph.png

\[\cos{x} = \frac{\text{adjacent}}{\text{hypotenuse}}\]

36 Cosine Graph.png

\[\tan{x} = \frac{\text{opposite}}{\text{adjacent}}\]

37 Tangent Graph.png

Using these trig ratios, we can solve problems involving distances, height, angles of elevation, angles of depression, etc.

Trigonometric Identities

Trigonometric identities are mathematical relationships established between the values of different trig functions. They are used to simplify expressions, solve equations, and prove mathematical theorems. Some common trig identities are shown below:

Pythagorean identities—These identities relate the squares of the trigonometric functions to each other:

38 Pythagorean Identities.png

sum and difference identities—These identities express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sines, cosines, and tangents of the individual angles:

39 Sum and Difference Identities.png

double angle identities—These identities express the sine, cosine, and tangent of double angles in terms of the sines, cosines, and tangents of the original angles:

40 Double Angle Identities.png