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Chapter 4: Problem 4

Express the given quantity using the same trigonometric ratio and its reference angle. For example, \(\cos 110^{\circ}=-\cos 70^{\circ}\) For angle measures in radians, give exact answers. For example, \(\cos 3=-\cos (\pi-3)\) a) \(\sin 250^{\circ}\) b) tan \(290^{\circ}\) c) sec \(135^{\circ}\) d) \(\cos 4\) e) csc 3 f) cot 4.95

Short Answer

Expert verified
a) \(\sin 250^{\circ} = -\sin 70^{\circ}\), b) \(\tan 290^{\circ} = -\tan 70^{\circ}\), c) \(\sec 135^{\circ} = -\sec 45^{\circ}\), d) \(\cos 4 = \cos (2\pi - 4)\), e) \(\csc 3 = \csc (\pi - 3)\), f) \(\cot 4.95 = -\cot (2\pi - 4.95)\).

Step by step solution

01

Identify Quadrant and Reference Angle (Part a)

For \(\sin 250^{\circ}\), determine the quadrant. Since \(\sin 250^{\circ}\) is in the third quadrant, the reference angle is \(\theta_r = 250^{\circ} - 180^{\circ} = 70^{\circ}\).
02

Apply Symmetry for Sine

In the third quadrant, sine is negative: \(\sin 250^{\circ} = -\sin 70^{\circ}\).
03

Identify Quadrant and Reference Angle (Part b)

For \(\tan 290^{\circ}\), determine the quadrant. Since \(\tan 290^{\circ}\) is in the fourth quadrant, the reference angle is \(\theta_r = 360^{\circ} - 290^{\circ} = 70^{\circ}\).
04

Apply Symmetry for Tangent

In the fourth quadrant, tangent is negative: \(\tan 290^{\circ} = -\tan 70^{\circ}\).
05

Identify Quadrant and Reference Angle (Part c)

For \(\sec 135^{\circ}\), determine the quadrant. Since \(\sec 135^{\circ}\) is in the second quadrant, the reference angle is \(\theta_r = 180^{\circ} - 135^{\circ} = 45^{\circ}\).
06

Apply Symmetry for Secant

In the second quadrant, secant is negative because cosine is negative: \(\sec 135^{\circ} = -\sec 45^{\circ}\).
07

Identify Quadrant and Reference Angle (Part d)

For \(\cos 4\), note that this angle is in radians. Since \(\cos 4\) is in the fourth quadrant, the reference angle is \(\theta_r = 2\pi - 4\).
08

Apply Symmetry for Cosine

In the fourth quadrant, cosine is positive: \(\cos 4 = \cos (2\pi - 4)\).
09

Identify Quadrant and Reference Angle (Part e)

For \(\csc 3\), note that this angle is in radians. Since \(\csc 3\) is in the second quadrant, the reference angle is \(\theta_r = \pi - 3\).
10

Apply Symmetry for Cosecant

In the second quadrant, cosecant is positive because sine is positive: \(\csc 3 = \csc (\pi - 3)\).
11

Identify Quadrant and Reference Angle (Part f)

For \(\cot 4.95\), note that this angle is in radians. Since \(\cot 4.95\) is in the fourth quadrant, the reference angle is \(\theta_r = 2\pi - 4.95\).
12

Apply Symmetry for Cotangent

In the fourth quadrant, cotangent is negative: \(\cot 4.95 = -\cot (2\pi - 4.95)\).

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Key Concepts

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

Reference Angles
A reference angle is the smallest angle made by the terminal side of the given angle and the horizontal axis.
Think of it as the angle's reflection within the first quadrant.
To calculate a reference angle, use these rules based on the given angle's quadrant:
  • Quadrant I: reference angle is equal to the given angle.
  • Quadrant II: subtract the given angle from 180掳 or 蟺.
  • Quadrant III: subtract 180掳 or 蟺 from the given angle.
  • Quadrant IV: subtract the given angle from 360掳 or 2蟺.
The Four Quadrants
The coordinate plane is divided into four sections called quadrants:
  • Quadrant I: Both x and y are positive.
  • Quadrant II: x is negative, y is positive.
  • Quadrant III: Both x and y are negative.
  • Quadrant IV: x is positive, y is negative.
Knowing which quadrant an angle falls into helps determine whether the trigonometric functions (sine, cosine, tangent, etc.) are positive or negative. For example, sine and cosine have different signs depending on which quadrant the angle is in.
Trigonometric Symmetry
Trigonometric functions such as sine, cosine, and tangent follow symmetrical properties depending on the angle's quadrant:
  • Sine is positive in Quadrants I and II; negative in Quadrants III and IV.
  • Cosine is positive in Quadrants I and IV; negative in Quadrants II and III.
  • Tangent is positive in Quadrants I and III; negative in Quadrants II and IV.
These properties are useful to match trigonometric functions of any angle with their reference angle while applying the correct sign.
Understanding Radians
Radians are an alternative way to measure angles, different from degrees. One full circle is equal to 2蟺 radians.
Here's a quick conversion:
  • 180掳 = 蟺 radians.
  • 90掳 = 蟺/2 radians.
  • 360掳 = 2蟺 radians.
When working with radians, it鈥檚 important to use the same trigonometric symmetry rules and to ensure angles fall within the specified quadrant for correct reference angle determination.
Negative and Positive Values
Trigonometric functions can have positive or negative values depending on the angle's position within the quadrants:
  • Positive values are found in the quadrants where both coordinates or their relevant trigonometric functions (like sine and cosine) are positive.
  • Negative values appear in quadrants where at least one of the relevant coordinates (or trigonometric functions) is negative.
Remember that reference angles are always positive, but you must apply the correct sign based on the angle鈥檚 quadrant.

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Most popular questions from this chapter

The equation \(\cos \theta=\frac{1}{2}, 0 \leq \theta < 2 \pi,\) has solutions \(\frac{\pi}{3}\) and \(\frac{5 \pi}{3} .\) Suppose the domain is not restricted. a) What is the general solution corresponding to \(\theta=\frac{\pi}{3} ?\) b) What is the general solution corresponding to \(\theta=\frac{5 \pi}{3} ?\)

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Nora is required to solve the following trigonometric equation. \(9 \sin ^{2} \theta+12 \sin \theta+4=0, \theta \in\left[0^{\circ}, 360^{\circ}\right)\) Nora did the work shown below. Examine her work carefully. Identify any errors. Rewrite the solution, making any changes necessary for it to be correct. \(9 \sin ^{2} \theta+12 \sin \theta+4=0\) \((3 \sin \theta+2)^{2}=0\) $$ 3 \sin \theta+2=0 $$ Therefore. \(\sin \theta=-\frac{2}{3}\) Use a colculator. \(\sin ^{-1}\left(-\frac{2}{3}\right)=-41.8103149\) So, the reference ongle is 41.8 , to the neorest tenth of a degree Sine is negotive in quodrants II ond III. The solution in quadront II is \(180^{\circ}-41.8^{\circ}=138.2\) The solution in quadrant III is \(180^{\circ}+41.8=221.8\) Therefore, \(\theta=138.2^{\circ}\) ond \(\theta=221.8\), to the neorest tenth of a degree.

Aslan and Shelley are finding the solution for \(2 \sin ^{2} \theta=\sin \theta, 0 < \theta \leq \pi\) Here is their work. \(2 \sin ^{2} \theta=\sin \theta\) \(\frac{2 \sin ^{2} \theta}{\sin \theta}=\frac{\sin \theta}{\sin \theta} \quad\) Step 1 \(2 \sin \theta=1 \quad\) Step 2 \(\sin \theta=\frac{1}{2} \quad\) Step 3 \(\theta=\frac{\pi}{6}, \frac{5 \pi}{6} \quad\) Step 4 a) Identify the error that Aslan and Shelley made and explain why their solution is incorrect. b) Show a correct method to determine the solution for \(2 \sin ^{2} \theta=\sin \theta, 0 < \theta \leq \pi\)

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