Problem 17 - Precalculus: Functions and Graphs

Question

Use a formula for negatives to find the exact value. $$\text { (a) } \sin \left(-90^{\circ}\right) \quad \text { (b) } \cos \left(-\frac{3 \pi}{4}\right) \quad \text { (c) } \tan \left(-45^{\circ}\right)$$

Step-by-Step Solution

Answer

(a) -1, (b) -√2/2, (c) -1

Step 1: Understanding Sine with Negative Angles

The formula for sine with a negative angle is given by \( \sin(-x) = - \sin(x) \). This tells us that the sine of a negative angle is the negative of the sine of the positive angle. For \( \sin(-90^\circ) \), this means \( \sin(-90^\circ) = -\sin(90^\circ) \).

Step 2: Calculating \( \sin(90^\circ) \)

From the unit circle, we know that the sine of \( 90^\circ \) is \( 1 \). Thus, \( \sin(-90^\circ) = -1 \times 1 = -1 \).

Step 3: Understanding Cosine with Negative Angles

The formula for cosine with a negative angle is \( \cos(-x) = \cos(x) \). This tells us that the cosine of a negative angle is the same as the cosine of the positive angle. For \( \cos\left(-\frac{3\pi}{4}\right) \), this means \( \cos\left(-\frac{3\pi}{4}\right) = \cos\left(\frac{3\pi}{4}\right) \).

Step 4: Calculating \( \cos\left(\frac{3\pi}{4}\right) \)

The angle \( \frac{3\pi}{4} \) radians corresponds to \( 135^\circ \). At this angle, cosine is \( -\frac{\sqrt{2}}{2} \), so \( \cos\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).

Step 5: Understanding Tangent with Negative Angles

The formula for tangent with a negative angle is \( \tan(-x) = -\tan(x) \). This means that the tangent of a negative angle is the negative of the tangent of the positive angle. For \( \tan(-45^\circ) \), this means \( \tan(-45^\circ) = -\tan(45^\circ) \).

Step 6: Calculating \( \tan(45^\circ) \)

From standard trigonometric values, we know that \( \tan(45^\circ) = 1 \). Therefore, \( \tan(-45^\circ) = -1 \times 1 = -1 \).

Key Concepts

Sine Function

The sine function is a fundamental component of trigonometry. It relates an angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. The formula for the sine of a negative angle is expressed as \( \sin(-x) = -\sin(x) \).
This identity tells us that the sine of a negative angle is simply the negative of the sine of its positive equivalent.
Let's apply this to calculate \( \sin(-90^\circ) \):

• The task involves calculating \( \sin(90^\circ) \).
• From the unit circle, \( \sin(90^\circ) \) is known to be equal to \( 1 \).
• Therefore, \( \sin(-90^\circ) = -1 \times 1 = -1 \).

Understanding how inverse angles behave is crucial in making accurate trigonometric calculations. This approach is valuable for any test or practical applications where these computations are necessary.

Cosine Function

The cosine function, another key trigonometric concept, describes the ratio of the adjacent side of a right triangle to the hypotenuse. What's particularly interesting about the cosine function is how it deals with negative angles. The identity for cosine is \( \cos(-x) = \cos(x) \), which indicates that cosine values are unaffected by the sign of the angle.
For example:

• To find \( \cos\left(-\frac{3\pi}{4}\right) \), we first calculate \( \cos\left(\frac{3\pi}{4}\right) \).
• The angle \( \frac{3\pi}{4} \) is equivalent to \( 135^\circ \), where cosine is \( -\frac{\sqrt{2}}{2} \).
• Therefore, \( \cos\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).

This consistency in cosine function values with the change in sign is helpful in scenarios dealing with symmetry and periodic phenomena in trigonometry.

Tangent Function

The tangent function offers a different perspective, connecting the two other functions, sine and cosine. Tangent of an angle is calculated by taking the ratio of the sine to the cosine of that angle. For negative angles, the tangent formula states \( \tan(-x) = -\tan(x) \), which mirrors the behavior we saw with the sine function.
Let's explore this with an example:

• We need to find \( \tan(-45^\circ) \).
• First, find \( \tan(45^\circ) \), which is known to be \( 1 \).
• Applying the negative angle identity, \( \tan(-45^\circ) = -1 \times 1 = -1 \).

Understanding the relationships among sine, cosine, and tangent is crucial, as these identities allow for manipulating and solving a vast range of trigonometric problems efficiently.