Use trigonometric function values of quadrantal angles to evaluate each expression. 3 sec 180° ― 5 tan 360°

Verified step by step guidance

1. Identify the trigonometric functions involved: secant (sec) and tangent (tan).
2. Recall the definitions of the secant and tangent functions: \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
3. Evaluate \( \sec 180^\circ \): Since \( \cos 180^\circ = -1 \), \( \sec 180^\circ = \frac{1}{-1} = -1 \).
4. Evaluate \( \tan 360^\circ \): Since \( \sin 360^\circ = 0 \) and \( \cos 360^\circ = 1 \), \( \tan 360^\circ = \frac{0}{1} = 0 \).
5. Substitute the evaluated values into the expression: \( 3 \times (-1) - 5 \times 0 \).

Key Concepts

Quadrantal Angles

Quadrantal angles are angles that are multiples of 90 degrees, specifically 0°, 90°, 180°, and 270°. These angles lie on the axes of the Cartesian plane, and their trigonometric function values can be easily determined. For example, the sine and cosine of these angles take on specific values, such as sin(180°) = 0 and cos(180°) = -1.

Trigonometric Functions

Trigonometric functions, including sine, cosine, tangent, secant, and others, relate the angles of a triangle to the ratios of its sides. For quadrantal angles, these functions yield specific values: for instance, sec(180°) = -1 and tan(360°) = 0. Understanding these functions is essential for evaluating expressions involving angles.

Evaluating Trigonometric Expressions

Evaluating trigonometric expressions involves substituting known values of trigonometric functions into mathematical expressions. In the given expression, 3 sec 180° and 5 tan 360°, one must first find the values of sec(180°) and tan(360°) before performing the arithmetic operations. This process is crucial for simplifying and solving trigonometric problems.