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Chapter 4: Problem 60

Find each value of \(\theta\) in degrees \(\left(0^{\circ}<\theta<90^{\circ}\right)\) and radians \((0<\theta<\pi\) 2) without using a calculator. (a) \(\tan \theta=\sqrt{3}\) (b) \(\cos \theta=\frac{1}{2}\)

Short Answer

Expert verified
For \(\tan \theta=\sqrt{3}\), the answer is \(\theta = 60^{\circ}\) or \(\theta = \pi /3\) in radians. For \(\cos \theta=\frac{1}{2}\), the answer is \(\theta = 60^{\circ}\) or \(\theta = \pi /3\) in radians.

Step by step solution

01

Evaluate for \(\tan \theta=\sqrt{3}\)

Remembering the values of \(\tan\) for standard angles, it is known that \(\tan 60^{\circ}\) or \(\tan (\pi /3)\) equals \(\sqrt{3}\). Therefore, the value of \(\theta\) that satisfies the equation \(\tan \theta=\sqrt{3}\) is \(\theta = 60^{\circ}\) or \(\theta = \pi /3\) in radians.
02

Evaluate for \(\cos \theta=\frac{1}{2}\)

Now, recalling the values of \(\cos\) for standard angles, it is known that \(\cos 60^{\circ}\) or \(\cos (\pi /3)\) equals \(\frac{1}{2}\). Therefore, the value of \(\theta\) that satisfies the equation \(\cos \theta=\frac{1}{2}\) is \(\theta = 60^{\circ}\) or \(\theta = \pi /3\) in radians.

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Key Concepts

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

Tangent Function
The tangent function, denoted as \( \tan \theta \), is a fundamental trigonometric function. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. In essence, it connects the angle \( \theta \) to two sides of the triangle.
This function is significant because it helps us understand the relationship between the angle and side lengths.
Some key points about the tangent function include:
  • It can take any real number as an input, but it is undefined for angles where the cosine of the angle is zero.
  • For angles \( \theta \) in the first quadrant (between \(0^{\circ}\) and \(90^{\circ}\)), the values of \( \tan \theta \) are positive.
  • Standard angles have well-known tangent values, which help in solving problems without a calculator.
When \( \tan \theta = \sqrt{3} \), like in the problem, we recognize this from the standard angle \(60^{\circ}\), or \(\pi/3\) radians.
Cosine Function
The cosine function, written as \( \cos \theta \), is another critical trigonometric function. It represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
Here are some essentials about the cosine function:
  • Values of \( \cos \theta \) range from -1 to 1. These values depict how strongly or weakly \( \theta \) correlates with its horizontal component.
  • In the first quadrant, the \( \cos \theta \) values are always positive.
  • This function helps determine the horizontal projection of the angle, crucial in many physics and engineering applications.
Recognizing that \( \cos \theta = \frac{1}{2} \) gives us insights into standard angles, such as \(60^{\circ}\) and \(\pi/3\) radians in our exercise.
Radians and Degrees
Radians and degrees are two units for measuring angles, and understanding the conversion between them is vital in trigonometry.
  • Degrees are arguably the most familiar unit, with a full circle being \(360^{\circ}\).
  • Radians are often used in higher mathematics, with one full rotation (circle) equating to \(2\pi\) radians.
  • To convert from degrees to radians, we utilize the relationship \( \pi \text{ radians} = 180^{\circ} \).
For example, since \( \pi/3 \) radians is equivalent to \(60^{\circ}\), it's crucial to be comfortable with such conversions in trigonometric problems.
Standard Angles
Standard angles are specific angles in trigonometry with well-known trigonometric values. These include \(0^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\), or their radian equivalents \(0\), \(\pi/6\), \(\pi/4\), \(\pi/3\), and \(\pi/2\) respectively.
Understanding standard angles is beneficial because:
  • They provide us with straightforward reference points without using calculators.
  • These angles simplify many trigonometric calculations.
  • They serve as benchmarks for understanding trigonometric functions in various quadrants.
Recognizing these standard angles can help quickly solve angles like \( \tan \theta = \sqrt{3} \) or \( \cos \theta = \frac{1}{2} \), evolving students' intuition in tackling trigonometric problems.

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Most popular questions from this chapter

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