10.3 Trigonometry

Learning Objectives

After Chapter 10.3, you will be able to:

• Explain the appropriate way to orient vectors in vector addition
• Calculate the value of sine, cosine, or tangent for a given right triangle
• Recall the sine, cosine, and tangent values of key angles
  

Very little trigonometry is required for the MCAT, but a basic understanding of definitions and a strong knowledge of two special right triangles is essential for strong performance, especially on physics material.

Definitions and Relationships

For any given right triangle and angle, there are characteristic values of sine, cosine, and tangent that depend on the lengths of the legs of the triangle and of the hypotenuse, as shown in Figure 10.1.

Figure 10.1. Right Triangle and Sides

Sine is calculated as the ratio between the side opposite the angle of interest and the hypotenuse:

Equation 10.17

Cosine is calculated as the ratio between the side adjacent to the angle of interest and the hypotenuse:

Equation 10.18

Tangent is calculated as the ratio between the side opposite the angle of interest and the side adjacent to the angle of interest:

Equation 10.19

Mnemonic

Trigonometric ratios: SOH CAH TOA:

• Sine = Opposite ÷ Hypotenuse
• Cosine = Adjacent ÷ Hypotenuse
• Tangent = Opposite ÷ Adjacent

The values of both sine and cosine range from −1 to 1. The values of tangent, however, range from −∞ to ∞.

Each trigonometric function also has an inverse function: inverse sine (sin⁻¹ or arcsin), inverse cosine (cos⁻¹ or arccos), and inverse tangent (tan⁻¹ or arctan). These functions use the calculated value of sine, cosine, or tangent, and yield a numerical value for the angle of interest. For the triangle in Figure 10.1, Inverse trigonometric functions are most likely to appear in questions asking for the direction of a resultant in vector addition or subtraction.

Key Concept

Trigonometric functions are useful for splitting a vector into its components; inverse trigonometric functions are useful for determining the direction of a resultant from its components.

Common Values

On Test Day, you must know the values of sine, cosine, and tangent for all of the angles in the 30−60−90 and 45−45−90 special right triangles, either by memorization or by drawing the triangles. The two triangles are shown in Figure 10.2.

Figure 10.2. Special Right Triangles (a) 30−60−90; (b) 45−45−90.

Important values of the trigonometric ratios at these angles are shown in Table 10.3.

Table 10.3. Common Trigonometric Ratios on the MCAT

θ	sin θ	cos θ	tan θ
0°	0	1	0
30°
45°			1
60°
90°	1	0	undefined
180°	0	−1	0

MCAT Concept Check 10.3

Before you move on, assess your understanding of the material with these questions.

1. During vector addition, how is the angle of the resultant calculated?

2. How are sine, cosine, and tangent calculated when given the dimensions of a right triangle?

   • Sine:
     
   • Cosine:
     
   • Tangent:

3. True or False: Only angles in right triangles have characteristic values of the trigonometric functions.

4. At what angle between 0° and 180° does each trigonometric function have a value of 1?