Problem 60 - Contemporary Precalculus

Question

In Exercises \(55-60\), find the values of all six trigonometric functions at \(t\) if the given conditions are true. $$\csc t=8 \quad \text { and } \quad \cos t<0$$

Step-by-Step Solution

Answer

Question: Given that \(\csc t = 8\) and \(\cos t < 0\), find the values of all six trigonometric functions at \(t\). Answer: We have found all six trigonometric functions for \(t\): $$\sin t = \frac{1}{8},\quad \cos t = -\frac{3\sqrt{7}}{8},\quad \tan t = -\frac{1}{3\sqrt{7}},$$ $$\csc t = 8,\quad \sec t = -\frac{8}{3\sqrt{7}},\quad \cot t = -3\sqrt{7}.$$

Step 1: Find \(\sin t\)

Since we know that \(\csc t = 8\), we can find \(\sin t\) by finding the reciprocal of \(\csc t\). So, we have $$\sin t = \frac{1}{\csc t} = \frac{1}{8}.$$

Step 2: Find \(\cos t\)

Using the Pythagorean identity, we have $$\cos^2 t = 1 - \sin^2 t = 1 - \left(\frac{1}{8}\right)^2 = 1 - \frac{1}{64} = \frac{63}{64}.$$ Since \(\cos t < 0\), we choose the negative square root: $$\cos t = -\sqrt{\frac{63}{64}} = -\frac{3\sqrt{7}}{8}.$$

Step 3: Find \(\tan t\)

Using the definition of tangent, we have $$\tan t = \frac{\sin t}{\cos t} = \frac{\frac{1}{8}}{-\frac{3\sqrt{7}}{8}} = -\frac{1}{3\sqrt{7}}.$$

Step 4: Find \(\cot t\)

Cotangent is the reciprocal of tangent, so we have $$\cot t = \frac{1}{\tan t} = \frac{1}{-\frac{1}{3\sqrt{7}}} = -3\sqrt{7}.$$

Step 5: Find \(\sec t\)

Secant is the reciprocal of cosine, so we have $$\sec t = \frac{1}{\cos t} = \frac{1}{-\frac{3\sqrt{7}}{8}} = -\frac{8}{3\sqrt{7}}.$$

Step 6: Summarize the results

We have found all six trigonometric functions for \(t\): $$\sin t = \frac{1}{8},\quad \cos t = -\frac{3\sqrt{7}}{8},\quad \tan t = -\frac{1}{3\sqrt{7}},$$ $$\csc t = 8,\quad \sec t = -\frac{8}{3\sqrt{7}},\quad \cot t = -3\sqrt{7}.$$

Key Concepts

Reciprocal Identities

Reciprocal identities in trigonometry are simple yet powerful tools that allow us to relate different trigonometric functions to one another. A reciprocal function means the inverse, or "flipping," of a fraction. This concept is pivotal when dealing with functions like sine and cosecant, cosine and secant, as well as tangent and cotangent. For example, if you know that \(\csc t = 8\), then you can immediately find \(\sin t\) by taking the reciprocal, yielding \(\sin t = \frac{1}{8}\). Similarly, you apply this idea to find secant from cosine or cotangent from tangent. Such identities deeply assist in simplifying trigonometric calculations and solving equations effectively.

• \(\csc t = \frac{1}{\sin t}\)
• \(\sec t = \frac{1}{\cos t}\)
• \(\cot t = \frac{1}{\tan t}\)

Mastering these basic reciprocal identities is fundamental for deeper trigonometric understanding.

Pythagorean Identity

The Pythagorean identity is derived from the Pythagorean Theorem and is one of the cornerstones of trigonometric mathematics. It states that for any angle \(t\):
\[ \sin^2 t + \cos^2 t = 1 \]
This equation provides a critical relationship between the sine and cosine of an angle, helping us find one when the other is known. For instance, when you know \(\sin t = \frac{1}{8}\), by substituting into the Pythagorean identity, you can solve for \(\cos t\). If \(\cos t\) is negative, it indicates the angle exists in either the second or third quadrant according to the unit circle.

• Helps derive other identities like:\(\tan^2 t + 1 = \sec^2 t\)
• Useful for finding angles given minimal information.
• Links geometrical concepts to trigonometric operations.

Being comfortable with the Pythagorean identity and its variants makes solving trigonometric problems much easier.

Negative Cosine Value

When dealing with trigonometric problems, the sign of the cosine function indicates the position of the angle on the coordinate plane. \(\cos t < 0\) suggests that the terminal side of the angle \(t\) is in either the second or third quadrant. This knowledge helps solve for unknowns; especially when combined with trigonometric identities.
In our exercise, after calculating the positive and then addressing the sign, you insert \(\cos^2 t = \frac{63}{64}\). To satisfy the condition \(\cos t < 0\), you choose the negative square root, ending with \(\cos t = -\frac{3\sqrt{7}}{8}\).

• Determines which trigonometric identity to apply.
• Provides directional context for angle measure through quadrants.
• Helps ascertain the sign of secant (sec), given that \( \sec t = \frac{1}{\cos t} \).

Recognizing when a trigonometric function is negative guides the calculation strategy and ensures valid results.

Inverse Trigonometric Functions

Inverse trigonometric functions are inverse operations of the basic trigonometric functions, allowing us to find angle measures from given trigonometric values. Functions like \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\) take these ratios and return angles, thus reversing the original trigonometric processes.
While not directly used in this particular exercise, understanding inverse functions is essential. They help find the specific values of angles that correspond to a particular function value, like finding an angle \(t\) such that \(\sin t = x\). Here’s a brief look into each:

• \(\sin^{-1}x\) provides an angle whose sine is \(x\).
• \(\cos^{-1}x\) provides an angle whose cosine is \(x\).
• \(\tan^{-1}x\) provides an angle whose tangent is \(x\).

Inverse functions broaden the usage of trigonometry, especially in real-world applications needing specific angle determination.