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Chapter 6: Problem 69

Find the reference angle and the exact function value if they exist. $$\cos 90^{\circ}$$

Short Answer

Expert verified
The reference angle is \(90^{\text{circ}}\) and \(\text{cos}(90^{\text{circ}}) = 0\).

Step by step solution

01

Understand the Problem

The exercise asks to find the reference angle and the exact value for the trigonometric function \(\text{cos} \) at \(90^{\text{circ}} \).
02

Identify the Standard Position Angle

The angle given is \(90^{\text{circ}} \), which is in the first quadrant. It is also known as a right angle.
03

Determine the Reference Angle

Since \(90^{\text{circ}} \) is the angle itself and located on the y-axis, it is its own reference angle. Therefore, the reference angle is \(90^{\text{circ}} \).
04

Evaluate the Cosine Function

Recall that the cosine of \(90^{\text{circ}} \) is \(0 \). This is a standard trigonometric value. \[ \cos 90^{\circ} = 0 \]

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Key Concepts

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

Reference Angle
The reference angle is an important concept in trigonometry. It helps simplify the process of finding trigonometric values for any given angle.
The reference angle is always between 0 and 90 degrees, regardless of the original angle's quadrant.
To find a reference angle:
  • If the angle is in the first quadrant, the reference angle is the angle itself.
  • If the angle is in the second quadrant, subtract the angle from 180 degrees.
  • If the angle is in the third quadrant, subtract the angle from 180 degrees, and then take the absolute value.
  • If the angle is in the fourth quadrant, subtract the angle from 360 degrees.
In our example, the angle is 90 degrees and lies on the y-axis, meaning the reference angle is 90 degrees.
Trigonometric Values
Trigonometric values are essential for understanding angles and their relationships in mathematics. These values include sine, cosine, and tangent.
Each function measures different aspects of a right triangle.
  • The sine function measures the ratio of the opposite side to the hypotenuse.
  • The cosine function measures the ratio of the adjacent side to the hypotenuse.
  • The tangent function measures the ratio of the opposite side to the adjacent side.
For the angle of 90 degrees, it's important to remember the specific trigonometric values. Here, \( \text{cos}(90^\text{circ}) = 0 \). This is a well-known trigonometric value. Therefore, the exact value of the cosine of 90 degrees is 0.
Standard Position Angle
Standard position angles are angles measured from the positive x-axis. These angles can either be positive, measured counterclockwise, or negative, measured clockwise.
The standard position helps identify where the angle lies relative to the coordinate plane's quadrants.
  • First quadrant: 0 to 90 degrees.
  • Second quadrant: 90 to 180 degrees.
  • Third quadrant: 180 to 270 degrees.
  • Fourth quadrant: 270 to 360 degrees.
In our example, the angle is 90 degrees, placing it in the first quadrant and specifically on the y-axis. Understanding the standard position angle ensures proper calculations of reference angles and trigonometric values.

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Most popular questions from this chapter

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