Problem 24
Question
Find the values of the six trigonometric functions of \(\boldsymbol{\theta}\) with the given constraint. $$ \cos \theta=\frac{8}{17} , tan \theta<0 $$
Step-by-Step Solution
Verified Answer
\(\sin \theta = -\frac{15}{17}\), \(\cos \theta = \frac{8}{17}\), \(\tan \theta = -\frac{15}{8}\), \(\cot \theta = -\frac{8}{15}\), \(\sec \theta = \frac{17}{8}\), \(\csc \theta = -\frac{17}{15}\)
1Step 1: Determine the Quadrant of \(\theta\)
The cosine function is positive in the first and fourth quadrants, and tangent is negative in the second and fourth quadrants. Therefore, \(\theta\) is in the fourth quadrant where both \(\cos \theta > 0\) and \(\tan \theta < 0\).
2Step 2: Determine the Value of \(\sin \theta\)
The sine function in the fourth quadrant is negative. We can use the identity \(\sin^2 \theta = 1 - \cos^2 \theta\) to calculate the magnitude of \(\sin \theta\), and since \(\theta\) is in the fourth quadrant, \(\sin \theta\) becomes negative. So, \(\sin \theta = -\sqrt{1 - \left(\frac{8}{17}\right)^2} = -\frac{15}{17}\)
3Step 3: Determine the Remaining Trigonometric Functions
Once \(\sin \theta\) and \(\cos \theta\) are known, the remaining trigonometric functions can be determined as follows: \(\tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{15}{8}\), \(\cot \theta = \frac{1}{\tan \theta} = -\frac{8}{15}\), \(\sec \theta = \frac{1}{\cos \theta} = \frac{17}{8}\), and \(\csc \theta = \frac{1}{\sin \theta} = -\frac{17}{15}\).
Key Concepts
Trigonometric IdentitiesQuadrant DeterminationSine and Cosine Functions
Trigonometric Identities
Trigonometric identities are like the Swiss Army knife for mathematics enthusiasts—extremely versatile and handy in a multitude of situations. When dealing with trigonometry, these identities enable us to simplify complex expressions, solve equations, and even compute unknown values.
The most basic identities include the Pythagorean Identity, which relates the sine and cosine of an angle, stating that \( sin^2 \theta + cos^2 \theta = 1 \). It comes in handy, especially when we know one of the trigonometric values (like cosine in the exercise) and we need to find another (like sine). Other essential identities involve the reciprocal relations like \( sec \theta = \frac{1}{cos \theta} \) and \( csc \theta = \frac{1}{sin \theta} \) as well as the tangent and cotangent ones \( tan \theta = \frac{sin \theta}{cos \theta} \) and \( cot \theta = \frac{1}{tan \theta} \).
Our exercise calls upon these identities. Knowing that \( cos \theta \) is positive and \( tan \theta \) is negative, we cleverly utilize the Pythagorean Identity to find \( sin \theta \) and then chain-react through the other identities to figure out the remaining trigonometric functions.
The most basic identities include the Pythagorean Identity, which relates the sine and cosine of an angle, stating that \( sin^2 \theta + cos^2 \theta = 1 \). It comes in handy, especially when we know one of the trigonometric values (like cosine in the exercise) and we need to find another (like sine). Other essential identities involve the reciprocal relations like \( sec \theta = \frac{1}{cos \theta} \) and \( csc \theta = \frac{1}{sin \theta} \) as well as the tangent and cotangent ones \( tan \theta = \frac{sin \theta}{cos \theta} \) and \( cot \theta = \frac{1}{tan \theta} \).
Our exercise calls upon these identities. Knowing that \( cos \theta \) is positive and \( tan \theta \) is negative, we cleverly utilize the Pythagorean Identity to find \( sin \theta \) and then chain-react through the other identities to figure out the remaining trigonometric functions.
Quadrant Determination
Picture the coordinate plane, with its axes splitting the plane into four quarters or 'quadrants'. In trigonometry, quadrants help us understand the signs (positive or negative) of different trigonometric functions. Starting from the positive x-axis and moving counterclockwise, the first quadrant is where all trig functions are positive.
As we move to the second quadrant, sine stays positive, but cosine and tangent flip to negative. In the third, it's the opposite; sine and tangent are negative, while cosine is positive. Finally, in the fourth quadrant, where our angle \(\theta\) lies according to the given exercise, cosine is positive, and both sine and tangent are negative.
Such quadrant determination is crucial because it guides us through the puzzling forest of trigonometry. It's particularly useful when we need to decide the sign of functions like sine or tangent, based on values of other functions.
As we move to the second quadrant, sine stays positive, but cosine and tangent flip to negative. In the third, it's the opposite; sine and tangent are negative, while cosine is positive. Finally, in the fourth quadrant, where our angle \(\theta\) lies according to the given exercise, cosine is positive, and both sine and tangent are negative.
Such quadrant determination is crucial because it guides us through the puzzling forest of trigonometry. It's particularly useful when we need to decide the sign of functions like sine or tangent, based on values of other functions.
Sine and Cosine Functions
The sine and cosine functions are the heartthrobs of trigonometry—fundamental to the subject. They relate the angles of a right triangle to the lengths of its sides.
The cosine of an angle—represented as \( cos \theta \)—is the ratio of the adjacent side to the hypotenuse in a right triangle. In the context of the unit circle (another important trigonometric model), it's the x-coordinate of the point where the terminal side of the angle intersects the circle.
Similarly, sine—\( sin \theta \)—is the ratio of the opposite side to the hypotenuse in a right triangle or the y-coordinate on the unit circle. These two functions are the gateway to determining the values of all other trigonometric functions. So, in our exercise, knowing \( cos \theta \) lets us unlock the value of \( sin \theta \) due to their deep connection through the Pythagorean Identity, which laid the foundation to solve for the rest of the trigonometric functions.
The cosine of an angle—represented as \( cos \theta \)—is the ratio of the adjacent side to the hypotenuse in a right triangle. In the context of the unit circle (another important trigonometric model), it's the x-coordinate of the point where the terminal side of the angle intersects the circle.
Similarly, sine—\( sin \theta \)—is the ratio of the opposite side to the hypotenuse in a right triangle or the y-coordinate on the unit circle. These two functions are the gateway to determining the values of all other trigonometric functions. So, in our exercise, knowing \( cos \theta \) lets us unlock the value of \( sin \theta \) due to their deep connection through the Pythagorean Identity, which laid the foundation to solve for the rest of the trigonometric functions.
Other exercises in this chapter
Problem 24
Sketch the graph of the function. Include two full periods. $$ y=-2 \sec 4 x+2 $$
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Describe the relationship between the graphs of \(f\) and \(g\). Consider amplitude, period, and shifts. $$ \begin{array}{l} f(x)=\sin x \\ g(x)=\sin 3 x \end{a
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Evaluate (if possible) the sine, cosine, and tangent of the real number. $$ t=\frac{5 \pi}{3} $$
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Sketch each angle in standard position. (a) \(-\frac{7 \pi}{4}\) (b) \(\frac{5 \pi}{2}\)
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