Sine, Cosine, and Tangent - AP PreCalculus Study Guide

The Core Idea: Sine, Cosine, and Tangent

This topic introduces the foundational trigonometric functions: sine, cosine, and tangent. The core idea is to define these functions not as abstract concepts, but as specific, tangible ratios of side lengths within a right triangle. For any acute angle within a right triangle, the values of its sine, cosine, and tangent are determined by the relationships between the lengths of the side opposite the angle, the side adjacent to the angle, and the hypotenuse.

These definitions provide a way to connect the angles of a right triangle to the lengths of its sides. By establishing these ratios, we can analyze the geometric properties of triangles and lay the groundwork for understanding more complex periodic phenomena. The tangent function is further defined as the ratio of the sine to the cosine, establishing a direct relationship between the three functions.

Key Formulas & Rules

The definitions of sine, cosine, and tangent for an acute angle $\theta$ in a right triangle are based on the lengths of the three sides relative to that angle: the side opposite$\theta$, the side adjacent to $\theta$, and the hypotenuse (which is always opposite the right angle).

Sine

The sine of an acute angle $\theta$, denoted as $\sin (\theta )$, is the ratio of the length of the side opposite angle $\theta$ to the length of the hypotenuse.

$$\sin (\theta )=\frac{\text{Length of Opposite Side}}{\text{Length of Hypotenuse}}$$

Cosine

The cosine of an acute angle $\theta$, denoted as $\cos (\theta )$, is the ratio of the length of the side adjacent to angle $\theta$ to the length of the hypotenuse.

$$\cos (\theta )=\frac{\text{Length of Adjacent Side}}{\text{Length of Hypotenuse}}$$

Tangent

The tangent of an acute angle $\theta$, denoted as $\tan (\theta )$, is the ratio of the length of the side opposite angle $\theta$ to the length of the side adjacent to angle $\theta$.

$$\tan (\theta )=\frac{\text{Length of Opposite Side}}{\text{Length of Adjacent Side}}$$

Tangent Identity

The tangent of an angle $\theta$ is also equal to the ratio of its sine to its cosine. This is a fundamental identity connecting the three functions.

$$\tan (\theta )=\frac{\sin (\theta )}{\cos (\theta )}$$

This can be shown by substituting the ratio definitions:

$$\frac{\sin (\theta )}{\cos (\theta )}=\frac{\frac{\text{Opposite}}{\text{Hypotenuse}}}{\frac{\text{Adjacent}}{\text{Hypotenuse}}}=\frac{\text{Opposite}}{\text{Hypotenuse}}\cdot \frac{\text{Hypotenuse}}{\text{Adjacent}}=\frac{\text{Opposite}}{\text{Adjacent}}=\tan (\theta )$$

Understanding the Ratios in a Right Triangle

The critical concept to grasp is that the terms "opposite" and "adjacent" are relative to the specific acute angle you are considering. For any right triangle, there are two acute angles. The side that is opposite one acute angle is adjacent to the other. The hypotenuse, however, is always the side opposite the right angle and is the longest side of the triangle.

Consider a right triangle with vertices A, B, and C, where C is the right angle. The sides are typically labeled with lowercase letters corresponding to the opposite vertex: side a is opposite angle A, side b is opposite angle B, and side c is the hypotenuse opposite angle C.

• For Angle A:

  • The opposite side is a.

  • The adjacent side is b.

  • The hypotenuse is c.

  • Therefore, $\sin (A)=\frac{a}{c}$, $\cos (A)=\frac{b}{c}$, and $\tan (A)=\frac{a}{b}$.

• For Angle B:

  • The opposite side is b.

  • The adjacent side is a.

  • The hypotenuse is c.

  • Therefore, $\sin (B)=\frac{b}{c}$, $\cos (B)=\frac{a}{c}$, and $\tan (B)=\frac{b}{a}$.

These definitions only apply to acute angles ($\theta$ such that $0^{\circ }<\theta <90^{\circ }$) within the context of a right triangle. The value of a trigonometric function for a given angle is constant, regardless of the size of the right triangle containing it. If two right triangles both contain an angle $\theta$, they are similar, and the ratios of their corresponding sides will be equal.

Core Concepts & Rules

• Right Triangle Context: The definitions of sine, cosine, and tangent as side ratios are valid for acute angles within a right triangle.

• Sine Ratio: The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.

• Cosine Ratio: The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

• Tangent Ratio: The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

• Relativity of Sides: The identification of a side as "opposite" or "adjacent" depends entirely on which acute angle is being used as the reference.

• Hypotenuse: The hypotenuse is always the side opposite the right angle and is never considered the adjacent or opposite side in the trigonometric ratios.

• Tangent as a Ratio of Sine and Cosine: The tangent of an angle is equivalent to its sine divided by its cosine.

Step-by-Step Example 1: Calculating Ratios from a Given Triangle

Problem: Consider the right triangle below with side lengths 5, 12, and 13. Determine the sine, cosine, and tangent of angle $\theta$.