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Chapter 8: Problem 8

Express the given trigonometric function in terms of the same function of a positive acute angle. $$\cos 190^{\circ}, \tan 290^{\circ}$$

Short Answer

Expert verified
\(\cos 190^\circ = -\cos 10^\circ\), \(\tan 290^\circ = -\tan 70^\circ\).

Step by step solution

01

Determine the Quadrant for 190掳

The angle 190掳 lies in the third quadrant, as it is between 180掳 and 270掳. In the third quadrant, the cosine function is negative.
02

Reference Angle for 190掳

The reference angle for 190掳 can be found by subtracting it from 180掳. So, the reference angle is 190掳 - 180掳 = 10掳.
03

Express Cosine in Terms of Reference Angle

Since cosine is negative in the third quadrant, \(\cos 190^\circ = -\cos 10^\circ\).
04

Determine the Quadrant for 290掳

The angle 290掳 lies in the fourth quadrant, as it is between 270掳 and 360掳. In the fourth quadrant, the tangent function is negative.
05

Reference Angle for 290掳

The reference angle for 290掳 can be found by subtracting it from 360掳. So, the reference angle is 360掳 - 290掳 = 70掳.
06

Express Tangent in Terms of Reference Angle

Since tangent is negative in the fourth quadrant, \(\tan 290^\circ = -\tan 70^\circ\).

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Key Concepts

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

Reference Angle
The concept of a reference angle is extremely helpful in trigonometry. A reference angle is the smallest positive angle that a given angle makes with the x-axis. It helps simplify the evaluation of trigonometric functions by reducing the size of the angle.
To find the reference angle:
  • If the angle is in the first quadrant (0掳 to 90掳), the reference angle equals the original angle.
  • If the angle is in the second quadrant (90掳 to 180掳), subtract the angle from 180掳.
  • If the angle is in the third quadrant (180掳 to 270掳), subtract 180掳 from the angle.
  • If the angle is in the fourth quadrant (270掳 to 360掳), subtract the angle from 360掳.
Using this method simplifies problems like \( \cos 190^{\circ} \) by using the reference angle of 10掳.
Quadrant Identification
Identifying the correct quadrant of an angle is important because the sign of trigonometric functions changes depending on the quadrant. Angles are measured from the positive x-axis in an anti-clockwise direction.
Here's how the quadrants work:
  • First quadrant: between 0掳 and 90掳, all trig functions are positive.
  • Second quadrant: between 90掳 and 180掳, sine is positive, and cosine and tangent are negative.
  • Third quadrant: between 180掳 and 270掳, sine and cosine are negative, and tangent is positive.
  • Fourth quadrant: between 270掳 and 360掳, cosine is positive, and sine and tangent are negative.
For example, 190掳 is in the third quadrant where cosine is negative and tangent is positive.
Angle Conversion
Sometimes working with negative angles, or angles greater than 360掳, requires converting them into a form that's easier to understand. This often involves either adding or subtracting multiples of 360掳 to find an equivalent angle.
For example:
  • Negative angles can be converted by adding 360掳 until a positive angle is achieved.
  • Angles over 360掳 can be reduced by subtracting 360掳 until the angle falls within standard range, i.e., 0掳 to 360掳.
This process ensures that any angle can be situated within one complete rotation of a circle, making calculations simpler and more intuitive.
Negative Angles Concept
Negative angles are measured in the clockwise direction from the positive x-axis, which is the opposite of the positive angle direction.
Key points about negative angles include:
  • Negative angles are often converted to positive by adding 360掳.
  • Trigonometric functions for negative angles have specific properties, like \( \cos(-\theta) = \cos(\theta) \) and \( \sin(-\theta) = -\sin(\theta) \).
Understanding this concept allows you to tackle problems involving any angle direction, leading to consistent results across different problems.

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