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

Verified step by step guidance

1. Recall the definitions and values of trigonometric functions at quadrantal angles: 180° and 360° are on the x-axis of the unit circle, where sine and cosine take specific values.
2. Evaluate \( \sec 180^\circ \). Since \( \sec \theta = \frac{1}{\cos \theta} \), first find \( \cos 180^\circ \), then take its reciprocal.
3. Evaluate \( \tan 360^\circ \). Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), so find \( \sin 360^\circ \) and \( \cos 360^\circ \) and compute their ratio.
4. Multiply the value of \( \sec 180^\circ \) by 3, and multiply the value of \( \tan 360^\circ \) by 5, as indicated in the expression.
5. Subtract the product \( 5 \tan 360^\circ \) from \( 3 \sec 180^\circ \) to get the final expression value.

Key Concepts

Quadrantal Angles

Quadrantal angles are angles that lie on the x- or y-axis in the coordinate plane, typically 0°, 90°, 180°, 270°, and 360°. Their trigonometric function values are special and often involve 0, ±1, or undefined values, which simplifies calculations.

Secant Function (sec θ)

The secant function is the reciprocal of the cosine function, defined as sec θ = 1/cos θ. Evaluating secant at quadrantal angles requires knowing the cosine values at those angles, which can be 0 or ±1, affecting whether secant is defined or undefined.

Tangent Function (tan θ)

The tangent function is the ratio of sine to cosine, tan θ = sin θ / cos θ. At quadrantal angles, tangent values can be 0, undefined, or ±1, depending on the sine and cosine values, which is crucial for correctly evaluating expressions involving tan 360°.