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Chapter 6: Problem 62

Let \(P(x, y)\) denote the point where the terminal side of angle \(\boldsymbol{\theta}\) (in standard position) meets the unit circle (as in Figure 4). Use the given information to evaluate the six trigonometric functions of \(\theta\). \(P\) is in Quadrant I and \(x=3 / 5\)

Short Answer

Expert verified
The six trigonometric functions are: \( \sin \theta = \frac{4}{5}, \cos \theta = \frac{3}{5}, \tan \theta = \frac{4}{3}, \csc \theta = \frac{5}{4}, \sec \theta = \frac{5}{3}, \cot \theta = \frac{3}{4} \).

Step by step solution

01

Understanding the Unit Circle

The problem gives a point \( P(x, y) \) where the terminal side of angle \( \theta \) meets the unit circle. In the unit circle, the equation is \( x^2 + y^2 = 1 \). With \( x = \frac{3}{5} \), we use this to find \( y \).
02

Finding the y-coordinate

Substitute \( x = \frac{3}{5} \) into the unit circle equation: \( \left(\frac{3}{5}\right)^2 + y^2 = 1 \). This becomes \( \frac{9}{25} + y^2 = 1 \). Simplify to find \( y^2 = 1 - \frac{9}{25} = \frac{16}{25} \), so \( y = \pm\frac{4}{5} \).
03

Determining the Correct y-coordinate

Since the point \( P \) is in Quadrant I, both \( x \) and \( y \) should be positive. Therefore, \( y = \frac{4}{5} \).
04

Calculating Sine and Cosine

\( \sin \theta = y = \frac{4}{5} \) and \( \cos \theta = x = \frac{3}{5} \). These correspond to the y and x coordinates respectively on the unit circle.
05

Calculating Tangent

\( \tan \theta = \frac{y}{x} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \). Tangent is the ratio of the sine to the cosine.
06

Calculating Cosecant

\( \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{4}{5}} = \frac{5}{4} \). Cosecant is the reciprocal of sine.
07

Calculating Secant

\( \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \). Secant is the reciprocal of cosine.
08

Calculating Cotangent

\( \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{4}{3}} = \frac{3}{4} \). Cotangent is the reciprocal of tangent.

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Key Concepts

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

Unit Circle
The unit circle is a foundational concept in trigonometry. It is a circle with a radius of 1 centered at the origin of the coordinate plane. This simple structure allows for easy identification of trigonometric values based on angles. Each point
  • on the unit circle corresponds to an angle, \(\theta\), formed by the radius (hypotenuse) and the positive x-axis.
  • The coordinates of each point on the unit circle are \((x, y)\), where \(x = \cos \theta\) and \(y = \sin \theta\).
  • This makes the unit circle a powerful tool for visualizing and understanding trigonometric functions.
For our problem, \(P(x, y)\) is on the unit circle, obeying the equation \(x^2 + y^2 = 1\). This equation confirms that any point \((x, y)\) lies exactly on the unit circle as the sum of the squares of the coordinates equals one.
Quadrants
The coordinate plane is divided into four quadrants, each defined by a combination of positive and negative x and y values. These quadrants help in identifying the signs of trigonometric functions
  • 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.
In our exercise, point \( P \) resides in Quadrant I, where both the x-coordinate and y-coordinate are positive. This affects the choice of \(y\) as positive after solving for it in the unit circle equation, ensuring our calculations for trigonometric functions align correctly.
Sine and Cosine
Sine and cosine are fundamental trigonometric functions that can be directly derived from the unit circle coordinates
  • Sine (sin) is the y-coordinate of the point \((x, y)\) on the unit circle corresponding to angle \(\theta\). For \(P\), it's \(\sin \theta = \frac{4}{5}\).
  • Cosine (cos), on the other hand, is the x-coordinate, thus \(\cos \theta = \frac{3}{5}\) for our angle.
These values allow us to form triangles that adhere to the radius of the unit circle. Together, they are crucial for defining other trigonometric functions and solving broader mathematical problems involving angles and distances.
Tangent and Cotangent
Tangent and cotangent are trigonometric functions derived from sine and cosine
  • Tangent (tan) is defined as the ratio of the sine of an angle to the cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). In this case, that's \(\tan \theta = \frac{4/5}{3/5} = \frac{4}{3}\).
  • Cotangent (cot) is the reciprocal of tangent: \(\cot \theta = \frac{1}{\tan \theta} = \frac{3}{4}\).
These ratios are especially useful when you need to find the slope of a line through the origin and any point \((x, y)\) on the unit circle. Having these values offers insight into the angle's characteristics relative to its sine and cosine.
Secant and Cosecant
Secant and cosecant functions are related reciprocals to cosine and sine, respectively
  • Secant (sec) is the reciprocal of cosine: \(\sec \theta = \frac{1}{\cos \theta} = \frac{5}{3}\).
  • Cosecant (csc) is the reciprocal of sine: \(\csc \theta = \frac{1}{\sin \theta} = \frac{5}{4}\).
These functions are not as intuitive as sine and cosine but are useful in solving problems involving angles and their amplitudes, particularly when the sine or cosine is small and their reciprocals grow larger.Understanding these reciprocals provides a complete perspective in trigonometry, enabling analysis in various mathematical and real-world contexts.

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Most popular questions from this chapter

Use a calculator to evaluate \(\sec \theta, \csc \theta,\) and cot \(\theta\) for the given value of \(\theta .\) Round the answers to two decimal places. $$5.23$$

Let \(P(x, y)\) denote the point where the terminal side of angle \(\boldsymbol{\theta}\) (in standard position) meets the unit circle (as in Figure 4). Use the given information to evaluate the six trigonometric functions of \(\theta\). $$x=-7 / 25 \text { and } 180_{i}<\theta<270_{i}$$

Prove that the equations are identities. $$\frac{\cot ^{2} A-\tan ^{2} A}{(\cot A+\tan A)^{2}}=2 \cos ^{2} A-1$$

This exercise completes the discussion in the text concerning the use of parentheses in calculator work. (a) Use the unit circle definitions to briefly explain (in complete sentences) why \(\sin (\pi / 2)=1\) and \(\sin \pi=0\). (b) Set the calculator to the radian mode and enter the following sequence of keystrokes. $$\begin{array}{|c|c|c|c|c|} \hline \text { sin } & \pi & \div & 2 & \text { ENTER } \\ \hline \end{array}$$ Your calculator will show an output of \(0,\) which, as you know, is not the value of \(\sin (\pi / 2) .\) This is because, in the absence of parentheses, the calculator interprets the sequence of keystrokes \((\sin ) \quad \pi \quad(\div) \quad 2\) as follows: First compute sin \(\pi\), then divide the result by 2 That is, the calculator computes \(0 \div 2,\) which, of course, results in the 0 output. Conclusion: If you want the calculator to compute \(\sin (\pi / 2),\) you must use parentheses and enter the sequence of keystrokes

Use the given information to determine the remaining five trigonometric values. $$\csc \beta=-\sqrt{5}, \quad 270^{\circ}<\beta<360^{\circ}$$
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