Evaluate each expression. See Example 4.cot² 135° - sin 30° + 4 tan 45°

Verified step by step guidance

1. Step 1: Recall the trigonometric identities and values for the given angles. For example, \( \cot 135^\circ = -1 \), \( \sin 30^\circ = \frac{1}{2} \), and \( \tan 45^\circ = 1 \).
2. Step 2: Calculate \( \cot^2 135^\circ \) by squaring the value of \( \cot 135^\circ \). Since \( \cot 135^\circ = -1 \), then \( \cot^2 135^\circ = (-1)^2 = 1 \).
3. Step 3: Substitute the known values into the expression: \( \cot^2 135^\circ - \sin 30^\circ + 4 \tan 45^\circ \) becomes \( 1 - \frac{1}{2} + 4 \times 1 \).
4. Step 4: Simplify the expression by performing the arithmetic operations: first, calculate \( 4 \times 1 \), then combine all terms.
5. Step 5: Combine the results from the previous steps to simplify the expression to its final form.

Key Concepts

Cotangent Function

The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function. It is defined as cot(θ) = cos(θ) / sin(θ). For angles in the unit circle, cotangent can be evaluated using the coordinates of the corresponding point. For example, cot(135°) can be calculated using the sine and cosine values of 135°, which are both negative.

Sine Function

The sine function, denoted as sin(θ), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is also defined on the unit circle as the y-coordinate of a point corresponding to the angle θ. For instance, sin(30°) equals 1/2, which is a fundamental value in trigonometry often used in various calculations.

Tangent Function

The tangent function, denoted as tan(θ), is defined as the ratio of the sine to the cosine of an angle, or tan(θ) = sin(θ) / cos(θ). It can also be interpreted as the slope of the line formed by the angle in the unit circle. For example, tan(45°) equals 1, which simplifies calculations involving tangent in trigonometric expressions.