Problem 33 - Precalculus Essentials

Question

Find the exact value of each of the remaining trigonometric functions of \(\theta .\) $$\sec \theta=-3, \quad \tan \theta>0$$

Step-by-Step Solution

Answer

The trigonometric values are as follows: \( \cos \theta = -1/3 \), \( \tan \theta = \sqrt{8} \), \( \sin \theta = -\sqrt{8}/3 \), \( \csc \theta = -3/\sqrt{8} \), \( \cot \theta = 1/\sqrt{8} \).

Step 1: Determine the Cosine Value

Given that \( \sec \theta = -3 \), it is known that \( \sec \theta \) is the reciprocal of \( \cos \theta \). So, \( \cos \theta = 1/ \sec \theta = -1/3 \). Now we know the value of \( \cos \theta \).

Step 2: Determine the Tangent Value

We can use the Pythagorean identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \) to find \( \tan(\theta) \). Substitute the given \( \sec(\theta) \) into the equation, you will get \( 1 + \tan^2(\theta) = (-3)^2 \). Solving this equation, we have \( \tan^2(\theta)=8 \). Since we know \( \tan \theta > 0 \), we take the positive square root to get \( \tan(\theta) = \sqrt{8} \).

Step 3: Determine the Sine Value

Having \( \cos \theta = -1/3 \) and \( \tan \theta = \sqrt{8} \), we can calculate \( \sin \theta \) using the formula \( \tan(\theta) = \sin(\theta) / \cos(\theta) \). Thus, \( \sin(\theta) = \cos(\theta) * \tan(\theta) = -1/3 * \sqrt{8} = -\sqrt{8}/3 \).

Step 4: Determine the Rest of the Trigonometric Function Values

The cosecant \( \csc(\theta) \) is the reciprocal of \( \sin(\theta) \), so \( \csc(\theta) = -3/\sqrt{8} \). The cotangent \( \cot(\theta) \) is the reciprocal of \( \tan(\theta) \), so \( \cot(\theta) = 1/\sqrt{8} \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental equation in trigonometry that relates three main trigonometric functions: sine, cosine, and one of either tangent or secant. This identity comes in a few variations, but the one relevant to our exercise is:

• \(1 + \tan^2(\theta) = \sec^2(\theta)\)

This equation shows that if you know the value of secant, you can solve for tangent, and vice versa. In our example, knowing \(\sec \theta = -3\) allows us to find that \(\tan^2(\theta) = 8\), since we square \(-3\) to get \(9\), then subtract \(1\) from it.
Pythagorean identities are derived from the Pythagorean Theorem and are essential when solving trigonometric equations or verifying identities. They often serve as tools to transform equations into simpler solutions and are one of the reasons we describe trig functions as interrelated.

Reciprocal Identities

Reciprocal identities are a key concept in trigonometry that express the relationships of trigonometric functions to their reciprocal counterparts. Each reciprocal identity is defined by flipping the original function:

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

In our exercise, the secant of theta was given as \(-3\), which allowed us to find \(\cos(\theta)\) by flipping this value: \(\cos(\theta) = -\frac{1}{3}\). Similarly, knowing \(\tan(\theta) = \sqrt{8}\) helped us find \(\cot(\theta) = \frac{1}{\sqrt{8}}\).
Understanding reciprocal identities can simplify complex problems, as working with reciprocals can often make equations easier to manage.

Tangent

Tangent is one of the primary trigonometric functions, expressing the ratio of the sine and cosine of an angle. It can be defined in terms of the unit circle or right-angled triangles:

• \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)

In a unit circle, tangent represents the slope of the line connecting the origin to the point \((\cos(\theta), \sin(\theta))\).
For the exercise, given the conditions \(\tan \theta > 0\), we determined the positive value of tangent as \(\tan(\theta) = \sqrt{8}\) using the Pythagorean identity \(1 + \tan^2(\theta) = \sec^2(\theta)\). This helped us solve for the other trigonometric functions of the angle.

Sine

The sine function is a foundational trigonometric function, represented as

• \(\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\)

for a right triangle. It is also the y-coordinate of a point on the unit circle corresponding to an angle \(\theta\).
In the context of our problem, we found \(\sin(\theta)\) using the relation with tangent: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). With \(\cos(\theta) = -1/3\) and \(\tan(\theta) = \sqrt{8}\), we calculated \(\sin(\theta) = -\frac{\sqrt{8}}{3}\). Knowing sine helps predict behaviors of waves and cycles, which is why it's such an essential function.

Cosine

Cosine is another main trigonometric function that describes the adjacent side over hypotenuse in a right triangle:

• \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\)

On the unit circle, it represents the x-coordinate of the point formed by angle \(\theta\) with the positive x-axis.
In our task, \(\cos(\theta)\) was calculated using the reciprocal identity of secant: \(\cos(\theta) = \frac{1}{\sec(\theta)} = -\frac{1}{3}\). Determining cosine is often a starting point for finding other trigonometric functions, as it provides essential relationships and simplifies computation of angles and sides in various contexts.