Problem 64 Evaluate the sine, cosine, and t... [FREE SOLUTION]

Chapter 4: Problem 64

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$ -\frac{\pi}{2} $$

Short Answer

Expert verified

The sine of $-\frac{\pi}{2}$ is -1, the cosine of $-\frac{\pi}{2}$ is 0, and the tangent of $-\frac{\pi}{2}$ is undefined.

Step by step solution

Evaluate sine

The sine of $-\frac{\pi}{2}$ is -1. This is because when the angle is $-\frac{\pi}{2}$, the terminal side of the angle intersects the unit circle at the point (0, -1). The y-coordinate of this point is the value of the sine function, so $\sin(-\frac{\pi}{2}) = -1$.

Evaluate cosine

The cosine of an angle is defined as the x-coordinate where the terminal side of the angle intersects the unit circle. For $-\frac{\pi}{2}$, this point is (0, -1). The x-coordinate of this point is 0, so $\cos(-\frac{\pi}{2}) = 0$.

Evaluate tangent

The tangent of an angle is the ratio of the sine to the cosine. In this case, the sine is -1 and the cosine is 0. Thus the tangent is $-1/0$, but division by zero is undefined in mathematics, so $\tan(-\frac{\pi}{2})$ is undefined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Circle

The unit circle is a fundamental concept in trigonometry, serving as a tool to find the values of trigonometric functions for various angles. It is a circle with a radius of 1, centered at the origin of a coordinate plane. This makes it very handy for simplifying calculations, since the radius relates directly to the circle's properties.

The circle is defined by the equation $ x^2 + y^2 = 1 $. Every point (x, y) on the unit circle satisfies this equation.
Angles on the unit circle can be measured in radians, where one full rotation around the circle equals $ 2\pi $ radians.
The unit circle is divided into four quadrants, each representing a distinct range of angles. These quadrants help in determining the sign of trigonometric functions.

Understanding how angles intersect the unit circle aids in finding sine, cosine, and tangent values without complicated calculations.

Sine Function

The sine function is based on the y-coordinate of a point on the unit circle. When you select an angle, the corresponding point where the terminal side intersects the unit circle provides the value of the sine function.

For the angle $ -\frac{\pi}{2} $, the terminal side reaches the point $ (0, -1) $ on the unit circle. Here, the y-coordinate is $-1$, thus $ \sin(-\frac{\pi}{2}) = -1 $.
The sine function is periodic, repeating every $ 2\pi $ radians. This periodicity helps in predicting the sine value for angles beyond the typical $ 0 $ to $ 2\pi $ range.

Key to mastering the sine function is appreciating its relationship with the unit circle and recognizing that it primarily concerns the vertical movement on this circle.

Cosine Function

The cosine function is another primary trigonometric function directly related to the unit circle. It is determined by the x-coordinate of the point where the terminal side of an angle intersects the circle.

For the angle $ -\frac{\pi}{2} $, the intersection point is $ (0, -1) $. The x-coordinate here is \ 0\, leading to $ \cos(-\frac{\pi}{2}) = 0 $.
Cosine, like sine, is periodic with a period of $ 2\pi $. This means the cosine value repeats every full circle.

In essence, the cosine function measures the horizontal projection of an angle's terminal side as it passes through the unit circle.

Tangent Function

The tangent function offers a different perspective on trigonometry, as it can be seen as the ratio of two other functions: sine and cosine. Specifically, the tangent of an angle is defined as the sine value divided by the cosine value.

Taking the example angle $ -\frac{\pi}{2} $, $ \tan(-\frac{\pi}{2}) = \frac{-1}{0} $. Since division by zero is undefined, tangent for this angle is not valid.
Tangent functions have asymptotes where the cosine is zero, which often results in undefined values.
Unlike sine and cosine, the tangent function has a periodicity of $ \pi $, repeating every half circle.

This understanding of the tangent function highlights its importance in trigonometry, particularly when interpreting the slopes of angles through the unit circle.

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91影视

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$ -\frac{\pi}{2} $$

Short Answer

Step by step solution

Evaluate sine

Evaluate cosine

Evaluate tangent

Key Concepts

Unit Circle

Sine Function

Cosine Function

Tangent Function

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