/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 Find the reference angle of each... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 55

Find the reference angle of each angle. $$ 440^{\circ} $$

Short Answer

Expert verified
80掳

Step by step solution

01

Understand Reference Angle

The reference angle is the smallest angle between the given angle and the x-axis. It is always between 0掳 and 90掳.
02

Reduce the Angle

To find the reference angle for an angle greater than 360掳, subtract 360掳 to bring the angle within one full rotation. For 440掳: \( 440^{\text{\textdegree}} - 360^{\text{\textdegree}} = 80^{\text{\textdegree}} \)
03

Determine the Reference Angle

Since the reduced angle from Step 2 is already within 0掳 and 90掳, the reference angle is simply \( 80^{\text{\textdegree}} \).

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Key Concepts

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

angle reduction
When working with angles in trigonometry, sometimes we encounter angles that are larger than 360掳. This means they have completed more than one full rotation around a circle.

To simplify these larger angles, we use a process called angle reduction. The goal is to reduce the angle until it falls within the first full rotation, between 0掳 and 360掳. We achieve this by subtracting 360掳 for every complete rotation the angle has made.

For example, in the exercise given with a 440掳 angle:
  • Subtract 360掳 from 440掳
  • This gives us 440掳 - 360掳 = 80掳
Now the angle is reduced to 80掳, which is within the first full rotation.
trigonometric angles
Trigonometric functions (sine, cosine, tangent, etc.) often deal with angles measured from the x-axis. These angles measure the rotation from the initial side (positive x-axis) to the terminal side of the angle.

These angles can be positive or negative:
  • Positive angles are measured counterclockwise from the x-axis.
  • Negative angles are measured clockwise from the x-axis.
When we talk about trigonometric angles, it鈥檚 important to understand how they repeat every 360掳. That鈥檚 why angle reduction is useful: it allows us to find an equivalent angle within the first 360掳 rotation, making it easier to work with trigonometric functions.
x-axis reference
The x-axis serves as a primary reference line in the coordinate system for measuring angles. The reference angle is the smallest angle between the terminal side of the given angle and the x-axis. It always lies between 0掳 and 90掳.

This is because the reference angle helps to find the corresponding values of trigonometric functions using the unit circle.

For example:
  • If you have an angle of 80掳 (already reduced from 440掳), and it's in the first quadrant, the reference angle is 80掳 itself.
  • If it were in another quadrant, you would determine the smallest angle to the x-axis from the terminal side.
The reference angle simplifies complex trigonometric calculations by standardizing them to the first quadrant.

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Most popular questions from this chapter

Use a calculator to find the approximate value of each expression. Round the answer to two decimal places. $$ \cot 70^{\circ} $$

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\cos 35^{\circ} \sin 55^{\circ}+\cos 55^{\circ} \sin 35^{\circ}$$

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$1-\cos ^{2} 15^{\circ}-\cos ^{2} 75^{\circ}$$

Convert each angle to a decimal in degrees. Round your answer to two decimal places.\(40^{\circ} 10^{\prime} 25^{\prime \prime}\)

Convert each angle in degrees to radians. Express your answer in decimal form, rounded to two decimal places. \(17^{\circ}\)
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