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Chapter 6: Problem 17

Find the exact values of the trigonometric functions for the acute angle \(\theta\). $$\sin \theta=\frac{3}{5}$$

Short Answer

Expert verified
\( \sin \theta = \frac{3}{5} \), \( \cos \theta = \frac{4}{5} \), \( \tan \theta = \frac{3}{4} \).

Step by step solution

01

Recall the Pythagorean Identity

The Pythagorean identity states that for any angle \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \). We will use this identity to find \( \cos \theta \) given \( \sin \theta = \frac{3}{5} \).
02

Substitute the Sin Value

Substitute \( \sin \theta = \frac{3}{5} \) into the Pythagorean Identity: \( \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \).
03

Simplify the Equation

Calculate the square of \( \frac{3}{5} \), which gives \( \frac{9}{25} \). Substitute back into the equation to get \( \frac{9}{25} + \cos^2 \theta = 1 \).
04

Solve for \( \cos^2 \theta \)

Subtract \( \frac{9}{25} \) from 1 to isolate \( \cos^2 \theta \). This results in \( \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} \).
05

Find \( \cos \theta \)

Take the square root of both sides to find \( \cos \theta \). Since \( \theta \) is an acute angle, \( \cos \theta \) is positive. Thus, \( \cos \theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \).
06

Find \( \tan \theta \)

To find \( \tan \theta \), use the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute \( \sin \theta = \frac{3}{5} \) and \( \cos \theta = \frac{4}{5} \) into this formula to get \( \tan \theta = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \).

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Key Concepts

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

Pythagorean identity
The Pythagorean identity is a fundamental concept in trigonometry. It states that for any angle \( \theta \), the sum of the squares of the sine and cosine of that angle equals one. Mathematically, this can be expressed as:
  • \( \sin^2 \theta + \cos^2 \theta = 1 \)
This identity is named after the Pythagorean theorem, due to the similarity between the relationships. It is an essential tool in trigonometry, providing a connection between sine and cosine. It is very useful when you know either \( \sin \theta \) or \( \cos \theta \) and need to find the other. For this exercise, knowing \( \sin \theta = \frac{3}{5} \) allows us to apply the Pythagorean identity to solve for \( \cos \theta \).
sin theta
The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse. In mathematical terms, if you have an angle \( \theta \), then:
  • \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
In our exercise, \( \sin \theta = \frac{3}{5} \). This means that if you consider a right triangle where the hypotenuse is 5 units long, the length of the side opposite the angle \( \theta \) is 3 units. Understanding sine is crucial for solving various problems in trigonometry, as it relates the angle with the sides of the triangle.
cos theta
The cosine function relates an angle \( \theta \) to the adjacent side and hypotenuse of a right triangle. It is defined as:
  • \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
This is particularly helpful when calculating angles or side lengths in triangles. In the exercise, the value of \( \cos \theta \) is derived using the Pythagorean identity, given \( \sin \theta = \frac{3}{5} \). Next, we solve for \( \cos \theta \) by rearranging the identity to \( \cos^2 \theta = 1 - \sin^2 \theta \). This results in \( \cos^2 \theta = \frac{16}{25} \), and taking the positive square root due to the acute angle, we find \( \cos \theta = \frac{4}{5} \).
tan theta
The tangent of an angle \( \theta \) is another essential trigonometric function, representing the ratio of the opposite side to the adjacent side of a right triangle. It can also be expressed in terms of sine and cosine as:
  • \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
In the context of this exercise, we are given \( \sin \theta = \frac{3}{5} \) and have calculated \( \cos \theta = \frac{4}{5} \). With these values, we find \( \tan \theta = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \). Understanding tangent is vital for solving problems that involve right triangles and circular functions, as it connects the sine and cosine functions into one comprehensive ratio.

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Most popular questions from this chapter

Practice sketching the graph of the sine function, taking different units of length on the horizontal and vertical axes. Practice sketching graphs of the cosine and tangent functions in the same manner. Continue this practice until you reach the stage at which, if you were awakened from a sound sleep in the middle of the night and asked to sketch one of these graphs, you could do so in less than thirty seconds.

Exer. 5-8: Let \(P(t)\) be the point on the unit circle \(U\) that corresponds to \(t\). If \(P(t)\) has the given rectangular coordinates, find (a) \(P(t+\pi)\) (b) \(P(t-\pi)\) (c) \(P(-t)\) (d) \(P(-t-\pi)\) $$ \left(-\frac{8}{17}, \frac{15}{17}\right) $$

Trigonometric functions are used extensively in the design of industrial robots. Suppose that a robot's shoulder joint is motorized so that the angle \(\theta\) increases at a constant rate of \(\pi / 12\) radian per second from an initial angle of \(\theta=0\). Assume that the elbow joint is always kept straight and that the arm has a constant length of 153 centimeters, as shown in the figure. (a) Assume that \(h=50 \mathrm{~cm}\) when \(\theta=0\). Construct a table that lists the angle \(\theta\) and the height \(h\) of the robotic hand every second while \(0 \leq \theta \leq \pi / 2\). (b) Determine whether or not a constant increase in the angle \(\theta\) produces a constant increase in the height of the hand. (c) Find the total distance that the hand moves.

Exer. 9-16: Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ}\), approximate the remaining parts. $$ a=0.42, \quad c=0.68 $$

Exer. 9-16: Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ}\), approximate the remaining parts. $$ a=31, \quad b=9.0 $$
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