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Chapter 6: Problem 80

Find the signs of the six trigonometric function values for the given angles. $$91^{\circ}$$

Short Answer

Expert verified
In \(91^{\circ}\), sin and csc are positive; cos, sec, tan, and cot are negative.

Step by step solution

01

Determine the quadrant

First, determine which quadrant the angle falls into. An angle of \(91^{\circ}\) is slightly greater than \(90^{\circ}\), placing it in the second quadrant.
02

Determine the signs of sine and cosine

In the second quadrant, the sine function is positive while the cosine function is negative.
03

Evaluate tangent

The tangent function, being the ratio of sine to cosine, will be negative in the second quadrant since it is positive divided by negative.
04

Evaluate cosecant

The cosecant function is the reciprocal of sine. Since sine is positive in the second quadrant, cosecant is also positive.
05

Evaluate secant

The secant function is the reciprocal of cosine. Because cosine is negative in the second quadrant, secant will also be negative.
06

Evaluate cotangent

The cotangent function is the reciprocal of tangent. Since tangent is negative in the second quadrant, cotangent will also be negative.

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Key Concepts

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

Quadrants
Understanding quadrants is crucial in trigonometry. Think of the coordinate system divided into four areas:
An angle in standard position starts at the positive x-axis. Here are the quadrants:
  • First Quadrant: angles from 0掳 to 90掳
  • Second Quadrant: angles from 90掳 to 180掳
  • Third Quadrant: angles from 180掳 to 270掳
  • Fourth Quadrant: angles from 270掳 to 360掳
An angle of 91掳 is slightly more than 90掳, which places it in the second quadrant.
Sine
The sine function represents the y-coordinate in the unit circle.
In the second quadrant, sine values are positive. This is because the y-coordinates in the second quadrant are above the x-axis.
For an angle (\theta) of 91掳, since it is in the second quadrant, \(\text{sin}(91掳) > 0\).
Cosine
The cosine function represents the x-coordinate in the unit circle.
In the second quadrant, cosine values are negative. This is because the x-coordinates in the second quadrant are to the left of the y-axis.
So for an angle (\theta) of 91掳, since it is in the second quadrant, \(\text{cos}(91掳) < 0\).
Tangent
Tangent is the ratio of sine to cosine: \(\text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)}\)
In the second quadrant, sine is positive and cosine is negative. Therefore, tangent will be negative.
For an angle (\theta) like 91掳, \(\text{tan}(91掳) < 0\).
Reciprocal Functions
Reciprocal trigonometric functions include cosecant, secant, and cotangent:
1. Cosecant (csc): Reciprocal of sine. Since sine is positive in the second quadrant, cosecant will also be positive.
2. Secant (sec): Reciprocal of cosine. Since cosine is negative in the second quadrant, secant will also be negative.
3. Cotangent (cot): Reciprocal of tangent. Since tangent is negative in the second quadrant, cotangent will also be negative.

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