Question

True or False Exact values can be found for the trigonometric functions of \(60^{\circ} .\)

Short answer

True, exact values can be found for the trigonometric functions of 60°.

Step-by-step solution

- Recall standard angles

Recall that there are certain angles, known as standard angles, for which trigonometric function values are often memorized due to their frequent use. These angles include 0°, 30°, 45°, 60°, and 90°.

- Identify if 60° is a standard angle

Identify from the list of standard angles whether 60° is among them. Note that 60° is indeed included in the list of common angles.

- Evaluate trigonometric functions for 60°

Find the exact values for the sine, cosine, and tangent functions of 60° using known formulas or a reference table. For 60°, we have the following exact values: \(\text{sin}(60°) = \frac{\sqrt{3}}{2}\) \(\text{cos}(60°) = \frac{1}{2}\) \(\text{tan}(60°) = \sqrt{3}\)

- Conclude based on findings

Since we were able to find the exact values for the sine, cosine, and tangent of 60°, we conclude that exact values can be found for the trigonometric functions of 60°.

Key concepts

exact values of trigonometric functions

In trigonometry, certain angles have specific trigonometric function values that are well-defined and can be precisely determined. These values are called exact values. When you hear about exact values, think of fractions or square roots rather than decimal approximations. For example, \( \text{sin}(60^{\text{°}}) = \frac{\text{√3}}{2} \), \( \text{cos}(60^{\text{°}}) = \frac{1}{2} \), and \( \text{tan}(60^{\text{°}}) = \text{√3} \). These values are exact because they are constant and can be derived using special triangles or the unit circle, rather than being subject to the inaccuracy of numerical approximation. Knowing these exact values is very useful, especially in more advanced math applications.

standard angles in trigonometry

Standard angles are the specific angles for which the trigonometric functions have well-known values. These commonly used angles include 0°, 30°, 45°, 60°, and 90°. Each of these angles is important because their trigonometric function values are easy to remember and often appear in various problems in trigonometry.
To understand why these angles are special, consider the 30°-60°-90° and 45°-45°-90° triangles that can be used to derive the exact values for these angles' trigonometric functions. For instance, in a 30°-60°-90° triangle:

• The side ratios are 1, √3, and 2,
• which help us find that \( \text{sin}(60^{\text{°}}) = \frac{\text{√3}}{2} \) and \( \text{cos}(60^{\text{°}}) = \frac{1}{2} \).

By memorizing these standard angles and their values, you simplify solving many trigonometric problems.

evaluating trigonometric functions

Evaluating trigonometric functions means finding the specific values of sin, cos, tan, and other trigonometric functions for a given angle. For standard angles, the evaluation process often involves recalling or deriving these values using special triangles or the unit circle.
Let's take 60° as an example:

• To find \( \text{sin}(60^{\text{°}}) \), you can use a reference table or internalize that it equals \( \frac{\text{√3}}{2} \).
• Similarly, \( \text{cos}(60^{\text{°}}) \) is \( \frac{1}{2} \), and \( \text{tan}(60^{\text{°}}) \) is \( \text{√3} \).

Knowing these values allows you to quickly and accurately solve trigonometry problems involving standard angles. Memorizing the exact values and understanding their derivation provides a solid foundation for more complex trigonometric concepts and applications.