Let Θ \theta Θ Be An Angle In Quadrant I Such That Sin ⁡ Θ = 4 5 \sin \theta = \frac{4}{5} Sin Θ = 5 4 ​ . Find The Exact Values Of Sec ⁡ Θ \sec \theta Sec Θ And Tan ⁡ Θ \tan \theta Tan Θ .

Introduction

In trigonometry, the sine, cosine, and tangent functions are used to describe the relationships between the angles and side lengths of triangles. Given an angle $\theta$ in quadrant I, we are often asked to find the exact values of trigonometric functions such as $\sin \theta$, $\cos \theta$, and $\tan \theta$. In this article, we will explore how to find the exact values of $\sec \theta$ and $\tan \theta$ when $\sin \theta = \frac{4}{5}$.

Understanding the Trigonometric Functions

Before we begin, let's review the definitions of the trigonometric functions. The sine, cosine, and tangent functions are defined as follows:

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

The secant function is the reciprocal of the cosine function, and the tangent function is the ratio of the opposite side to the adjacent side.

Finding the Exact Values of $\sec \theta$ and $\tan \theta$

Given that $\sin \theta = \frac{4}{5}$, we can use the Pythagorean identity to find the exact values of $\cos \theta$, $\sec \theta$, and $\tan \theta$.

Using the Pythagorean Identity

The Pythagorean identity states that $\sin^2 \theta + \cos^2 \theta = 1$. We can use this identity to find the exact value of $\cos \theta$.

import math
sin_theta = 4/5

cos_theta = math.sqrt(1 - sin_theta**2)

Finding the Exact Value of $\cos \theta$

Using the Pythagorean identity, we can calculate the exact value of $\cos \theta$.

# Calculate cos theta
cos_theta = math.sqrt(1 - (4/5)**2)
print(cos_theta)

Finding the Exact Value of $\sec \theta$

The secant function is the reciprocal of the cosine function. Therefore, we can find the exact value of $\sec \theta$ by taking the reciprocal of $\cos \theta$.

# Calculate sec theta
sec_theta = 1 / cos_theta
print(sec_theta)

Finding the Exact Value of $\tan \theta$

The tangent function is the ratio of the opposite side to the adjacent side. We can find the exact value of $\tan \theta$ by dividing the opposite side by the adjacent side.

# Calculate tan theta
tan_theta = sin_theta / cos_theta
print(tan_theta)

Conclusion

In this article, we explored how to find the exact values of $\sec \theta$ and $\tan \theta$ when $\sin \theta = \frac{4}{5}$. We used the Pythagorean identity to find the exact value of $\cos \theta$, and then used the reciprocal and ratio definitions of the secant and tangent functions to find their exact values. By following these steps, we can find the exact values of $\sec \theta$ and $\tan \theta$ for any given angle in quadrant I.

Final Answer

The final answer is:

• $\sec \theta = \frac{5}{3}$
• $\tan \theta = \frac{4}{3}$
  

Introduction

In our previous article, we explored how to find the exact values of $\sec \theta$ and $\tan \theta$ when $\sin \theta = \frac{4}{5}$. In this article, we will answer some common questions related to finding the exact values of trigonometric functions.

Q1: What is the Pythagorean identity?

A1: The Pythagorean identity is a fundamental concept in trigonometry that states $\sin^2 \theta + \cos^2 \theta = 1$. This identity can be used to find the exact values of trigonometric functions.

Q2: How do I find the exact value of $\cos \theta$ when $\sin \theta = \frac{4}{5}$?

A2: To find the exact value of $\cos \theta$, you can use the Pythagorean identity: $\cos^2 \theta = 1 - \sin^2 \theta$. Then, take the square root of both sides to find the exact value of $\cos \theta$.

Q3: What is the reciprocal of the cosine function?

A3: The reciprocal of the cosine function is the secant function, denoted by $\sec \theta$. Therefore, $\sec \theta = \frac{1}{\cos \theta}$.

Q4: How do I find the exact value of $\tan \theta$ when $\sin \theta = \frac{4}{5}$?

A4: To find the exact value of $\tan \theta$, you can divide the opposite side by the adjacent side: $\tan \theta = \frac{\sin \theta}{\cos \theta}$. Then, substitute the given value of $\sin \theta$ and the exact value of $\cos \theta$ to find the exact value of $\tan \theta$.

Q5: Can I use the Pythagorean identity to find the exact values of other trigonometric functions?

A5: Yes, the Pythagorean identity can be used to find the exact values of other trigonometric functions. For example, you can use the Pythagorean identity to find the exact values of $\sin \theta$ and $\tan \theta$ when $\cos \theta$ is given.

Q6: What is the relationship between the secant and tangent functions?

A6: The secant function is the reciprocal of the cosine function, and the tangent function is the ratio of the opposite side to the adjacent side. Therefore, $\sec \theta = \frac{1}{\cos \theta}$ and $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

Q7: How do I find the exact value of $\sec \theta$ when $\sin \theta = \frac{4}{5}$?

A7: To find the exact value of $\sec \theta$, you can take the reciprocal of the exact value of $\cos \theta$: $\sec \theta = \frac{1}{\cos \theta}$.

Q8: Can I use a calculator to find the exact values of trigonometric functions?

A8: Yes, you can use a calculator to find the exact values of trigonometric functions. However, it's always a good idea to verify the results using the Pythagorean identity and other trigonometric identities.

Conclusion

In this article, we answered some common questions related to finding the exact values of trigonometric functions. We hope that this article has been helpful in clarifying any doubts you may have had. Remember to always use the Pythagorean identity and other trigonometric identities to find the exact values of trigonometric functions.

Final Answer

The final answer is:

• $\sec \theta = \frac{5}{3}$
• $\tan \theta = \frac{4}{3}$