Problem 63 - Precalculus

Question

Evaluate the following expressions exactly: $$\tan \left(-315^{\circ}\right)$$

Step-by-Step Solution

Answer

\( \tan(-315^{\circ}) = 1 \).

Step 1: Convert Negative Angle to Positive

To find \( \tan(-315^{\circ}) \), first convert the negative angle to a positive angle. Add \( 360^{\circ} \) to \( -315^{\circ} \) to get \( -315^{\circ} + 360^{\circ} = 45^{\circ} \). This uses the periodicity of the tangent function, which has a period of \( 180^{\circ} \), to simplify the angle.

Step 2: Evaluate Tangent of the Positive Angle

Now that we have converted the angle to \( 45^{\circ} \), evaluate \( \tan(45^{\circ}) \). The tangent of \( 45^{\circ} \) is known to be 1. Therefore, \( \tan(45^{\circ}) = 1 \).

Key Concepts

Angle Conversion

Angle conversion is a fundamental concept when working with trigonometric functions. In trigonometry, angles can be measured in degrees or radians. Converting between negative and positive angles helps solve problems easily. When faced with a negative angle, we add a full rotation of \(360^{\circ}\) (or \(2\pi\) radians) to find the equivalent positive angle.
This process doesn't alter the trigonometric properties of the angle.

• Convert \(-315^{\circ}\) to a positive angle by adding \(360^{\circ}\).
• This results in the positive angle of \(45^{\circ}\).

This step ensures our calculations remain within the standard cycle of trigonometric functions, making it simpler to evaluate functions like tangent.

Periodicity of Tangent

The periodicity of the tangent function is a key property that simplifies complex angle evaluations. Unlike sine and cosine, with a period of \(360^{\circ}\) (or \(2\pi\) radians), tangent has a period of \(180^{\circ}\) (or \(\pi\) radians). This means the tangent function repeats its values every \(180^{\circ}\).

• For instance, \(\tan(\theta) = \tan(\theta + 180^{\circ})\).
• In our example, converting \(-315^{\circ}\) to \(45^{\circ}\) still results in the same tangent value due to this periodicity.

This property is particularly useful in converting angles and simplifying calculations, as different angles that differ by multiples of \(180^{\circ}\) will have identical tangent values.

Tangent of 45 Degrees

The tangent of \(45^{\circ}\) is a widely known and straightforward value in trigonometry.
The tangent function, defined as the ratio of the opposite side to the adjacent side in a right triangle, takes a special meaning at \(45^{\circ}\).
At this angle, both sides in a right-angled triangle are equal, making the tangent value 1.
Thus we have:

• \(\tan(45^{\circ}) = \frac{1}{1} = 1\).

Remembering these simple angles and their tangent values helps tremendously when evaluating or verifying trigonometric expressions, providing a quick reference to frequently used results.