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Chapter 7: Problem 56

Find the reference angle of each angle. $$ 490^{\circ} $$

Short Answer

Expert verified
The reference angle is 50掳.

Step by step solution

01

Understand reference angles

A reference angle is the smallest angle between the terminal side of the given angle and the x-axis. It is always measured in the positive direction.
02

Find the coterminal angle in the range [0^{\text{ \textdegree}}, 360^{\text{ \textdegree}}]

To find the coterminal angle, subtract multiples of 360掳 from 490掳 until the result is within the range [0掳, 360掳]. 490掳 - 360掳 = 130掳 .
03

Determine the reference angle

Since 130掳 is in the 2nd quadrant, the reference angle is calculated by subtracting the angle from 180掳: 180掳 - 130掳 = 50掳.

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Key Concepts

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

Coterminal Angles
Coterminal angles are angles that share the same terminal side when drawn in standard position. This means that even though they may have different measures, they essentially point in the same direction.
The simplest way to find a coterminal angle is by adding or subtracting multiples of 360掳 because a full circle is 360掳. For example, starting with an angle of 490掳:
  • 490掳 - 360掳 leads to 130掳
  • We could also add 360掳 to a negative angle, like -30掳, and get 330掳 because -30掳 + 360掳 = 330掳
The key is that adding or subtracting 360掳 from an angle gives you another angle that looks the same on a unit circle.
Quadrants
The coordinate plane is divided into four sections known as quadrants, each with specific characteristics. This system helps us identify the sign of the sine, cosine, and tangent functions based on the angle's position.
  • First Quadrant: Angles from 0掳 to 90掳, both sine and cosine are positive.
  • Second Quadrant: Angles from 90掳 to 180掳, sine is positive, cosine is negative.
  • Third Quadrant: Angles from 180掳 to 270掳, both sine and cosine are negative.
  • Fourth Quadrant: Angles from 270掳 to 360掳, sine is negative, cosine is positive.
In our example, 130掳 falls into the second quadrant, so we use this information to find the reference angle by subtracting from 180掳.
Angle Measurement
Angle measurement can be in degrees or radians, but degrees are more common for basic trigonometry. Angles in standard position are measured from the positive x-axis counterclockwise.
Here's a quick overview:
  • Full circle: 360掳
  • Straight angle: 180掳
  • Right angle: 90掳
When measuring angles greater than 360掳 or negative angles, we often use coterminal angles to simplify. For instance, the angle 490掳 can be thought of as equivalent to 130掳, by subtracting 360掳. This helps us work with simpler values while acknowledging the angle's true measure.

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Most popular questions from this chapter

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\frac{\cos 40^{\circ}}{\sin 50^{\circ}}$$

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Determine the amplitude and period of each function without graphing. $$ y=\frac{4}{3} \sin \left(\frac{2}{3} x\right) $$

The function \(y=-3 \cos (6 x)\) has amplitude ________ and period _______.

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