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Chapter 6: Problem 77

Find the signs of the six trigonometric function values for the given angles. $$-215^{\circ}$$

Short Answer

Expert verified
In the second quadrant, \(\sin\) and \(\csc\) are positive, \(\cos\), \(\sec\), \(\tan\), and \(\cot\) are negative.

Step by step solution

01

Convert the Angle

First, convert the given angle o to its equivalent positive angle. To do this, add \(360^{\circ}\) until the angle is positive: ooo\(-215^{ o}+360^{\circ}=145^{\circ}\).
02

Determine the Quadrant

Determine which quadrant the angle \(145^{\circ}\) is in. oSince it lies between \(90^{\circ}\) and \(180^{\circ}\), \(145^{\circ}\) is in the second quadrant.
03

Understand the Signs in Quadrants

Recall the signs of the trigonometric functions for angles in the second quadrant: In the second quadrant, sine and cosecant are positive, while cosine, secant, tangent, and cotangent are negative.
04

Assign the Appropriate Signs

- The signs of the trigonometric functions at \(-215^{\circ}\) are as follows:ooo\(\sin(-215^{\circ})\) is positive鈥⑩�o\(\cos(-215^{\circ})\) is negative鈥⑩�o\(\tan(-215^{\circ})\) is negative鈥⑩�o\(\csc(-215^{\circ})\) is positive鈥⑩�oo\(\sec(-215^{\circ})\) is negative鈥oo\(\cot(-215^{\circ})\) is negativeoo

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Key Concepts

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

Angle Conversion
Angle conversion is essential when dealing with negative angles or large positive angles in trigonometry. To convert a negative angle to its positive equivalent, we add multiples of 360 degrees until the value becomes positive. This is because the unit circle, which is fundamental in trigonometry, repeats every 360 degrees. Hence, for the angle \-215^{\circ}\, we add 360 degrees: \( -215^{\circ} + 360^{\circ} = 145^{\circ} \) Now, we have a positive angle of \( 145^{\circ} \) which is easier to work with when determining the signs of trigonometric functions.
Trigonometric Quadrants
The unit circle is divided into four quadrants, each with specific sign conventions for trigonometric functions. Angles from \(0^{\circ} \) to \( 90^{\circ} \) are in the first quadrant. Angles from\( 90^{\circ} \) to \(180^{\circ} \) are in the second quadrant. Angles from \( 180^{\circ} \) to \( 270^{\circ} \) are in the third quadrant. Angles from \( 270^{\circ} \) to \( 360^{\circ} \) are in the fourth quadrant. Since \( 145^{\circ} \) lies between \( 90^{\circ} \) and \(180^{\circ} \), it is in the second quadrant. Understanding which quadrant an angle falls into is important for identifying the signs of the trigonometric functions.
Trigonometric Function Signs
In each quadrant, different trigonometric functions have different signs.
  • In the first quadrant, all trigonometric functions are positive.
  • In the second quadrant, sine and cosecant are positive while cosine, secant, tangent, and cotangent are negative.
  • In the third quadrant, tangent and cotangent are positive while sine, cosine, secant, and cosecant are negative.
  • In the fourth quadrant, cosine and secant are positive while sine, tangent, cotangent, and cosecant are negative.

Thus, for an angle of \( -215^{\circ} \), which is equivalent to \( 145^{\circ} \) in the second quadrant:
  • \( \sin(-215^{\circ}) \) is positive
  • \( \cos(-215^{\circ}) \) is negative
  • \( \tan(-215^{\circ}) \) is negative

  • \( \csc(-215^{\circ}) \) is positive
  • \( \sec(-215^{\circ}) \) is negative
  • \( \cot(-215^{\circ}) \) is negative
Knowing these signs helps solve various trigonometric problems more efficiently.

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Most popular questions from this chapter

Given the function value and the quadrant restriction, find \(\theta\). FUNCTION VALUE = \(\cos \theta=-0.9388\) INTERVAL = \(\left(180^{\circ}, 270^{\circ}\right)\) \(\boldsymbol{\theta}\) = ____

The transformation techniques that we learned in this section for graphing the sine and cosine functions can also be applied to the other trigonometric functions. Sketch a graph of each of the following. Then check your work using a graphing calculator. $$y=4 \sec (2 x-\pi)$$

Classify the function as linear, quadratic, cubic, quartic, rational, exponential, logarithmic, or trigonometric. $$\frac{3}{4} x+\frac{1}{2} y=-5[1.3]$$

Find the function value. Round to four decimal places. $$\cot 146.15^{\circ}$$

The transformation techniques that we learned in this section for graphing the sine and cosine functions can also be applied to the other trigonometric functions. Sketch a graph of each of the following. Then check your work using a graphing calculator. $$y=-\tan x$$
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