Evaluate Without Using A Calculator. Use The Pythagorean Identities Rather Than Reference Triangles.Find $\tan \theta$ And $\cot \theta$ If $\sec \theta = \frac{8}{5}$ And $\sin \theta \ \textless \ 0$.$\tan

Introduction

In trigonometry, the Pythagorean identities are a set of fundamental relationships between the trigonometric functions. These identities can be used to express one trigonometric function in terms of another, without the need for reference triangles. In this article, we will use the Pythagorean identities to evaluate the tangent and cotangent of an angle, given the secant of the angle.

Pythagorean Identities

The Pythagorean identities are:

• $\sin^2 \theta + \cos^2 \theta = 1$
• $\tan^2 \theta + 1 = \sec^2 \theta$
• $\cot^2 \theta + 1 = \csc^2 \theta$

These identities can be used to express one trigonometric function in terms of another.

Evaluating $\tan \theta$ and $\cot \theta$

We are given that $\sec \theta = \frac{8}{5}$ and $\sin \theta < 0$. We can use the Pythagorean identities to evaluate the tangent and cotangent of the angle.

Evaluating $\tan \theta$

We can use the identity $\tan^2 \theta + 1 = \sec^2 \theta$ to evaluate the tangent of the angle.

$\tan^2 \theta + 1 = \sec^2 \theta$

$\tan^2 \theta + 1 = \left(\frac{8}{5}\right)^2$

$\tan^2 \theta + 1 = \frac{64}{25}$

$\tan^2 \theta = \frac{64}{25} - 1$

$\tan^2 \theta = \frac{64}{25} - \frac{25}{25}$

$\tan^2 \theta = \frac{39}{25}$

$\tan \theta = \pm \sqrt{\frac{39}{25}}$

Since $\sin \theta < 0$, we know that $\tan \theta < 0$. Therefore, we take the negative square root.

$\tan \theta = -\sqrt{\frac{39}{25}}$

$\tan \theta = -\frac{\sqrt{39}}{\sqrt{25}}$

$\tan \theta = -\frac{\sqrt{39}}{5}$

Evaluating $\cot \theta$

We can use the identity $\cot^2 \theta + 1 = \csc^2 \theta$ to evaluate the cotangent of the angle. However, we are not given the value of $\csc \theta$. Instead, we can use the identity $\cot \theta = \frac{1}{\tan \theta}$.

$\cot \theta = \frac{1}{\tan \theta}$

$\cot \theta = \frac{1}{-\frac{\sqrt{39}}{5}}$

$\cot \theta = -\frac{5}{\sqrt{39}}$

$\cot \theta = -\frac{5}{\sqrt{39}} \times \frac{\sqrt{39}}{\sqrt{39}}$

$\cot \theta = -\frac{5\sqrt{39}}{39}$

Conclusion

In this article, we used the Pythagorean identities to evaluate the tangent and cotangent of an angle, given the secant of the angle. We found that $\tan \theta = -\frac{\sqrt{39}}{5}$ and $\cot \theta = -\frac{5\sqrt{39}}{39}$.

References

• [1] "Trigonometry" by Michael Corral, 2015.
• [2] "Calculus" by Michael Spivak, 2008.

Further Reading

• For more information on the Pythagorean identities, see [1].
• For more information on trigonometry, see [2].

Code

import math

def evaluate_tan(theta):
    sec_theta = 8/5
    tan_squared_theta = (sec_theta**2) - 1
    tan_theta = -math.sqrt(tan_squared_theta)
    return tan_theta

def evaluate_cot(theta):
    tan_theta = evaluate_tan(theta)
    cot_theta = 1 / tan_theta
    return cot_theta

theta = math.acos(-3/5)
tan_theta = evaluate_tan(theta)
cot_theta = evaluate_cot(theta)

print("The value of tan(θ) is: ", tan_theta)
print("The value of cot(θ) is: ", cot_theta)

Note: The code above is for demonstration purposes only and is not intended to be used for actual calculations.

Introduction

In our previous article, we used the Pythagorean identities to evaluate the tangent and cotangent of an angle, given the secant of the angle. In this article, we will answer some frequently asked questions about evaluating trigonometric functions using Pythagorean identities.

Q: What are the Pythagorean identities?

A: The Pythagorean identities are a set of fundamental relationships between the trigonometric functions. They are:

• $\sin^2 \theta + \cos^2 \theta = 1$
• $\tan^2 \theta + 1 = \sec^2 \theta$
• $\cot^2 \theta + 1 = \csc^2 \theta$

Q: How can I use the Pythagorean identities to evaluate trigonometric functions?

A: You can use the Pythagorean identities to express one trigonometric function in terms of another. For example, if you are given the secant of an angle, you can use the identity $\tan^2 \theta + 1 = \sec^2 \theta$ to evaluate the tangent of the angle.

Q: What is the difference between the Pythagorean identities and reference triangles?

A: The Pythagorean identities are a set of mathematical relationships between the trigonometric functions, while reference triangles are geometric representations of the trigonometric functions. The Pythagorean identities can be used to evaluate trigonometric functions without the need for reference triangles.

Q: Can I use the Pythagorean identities to evaluate all trigonometric functions?

A: Yes, you can use the Pythagorean identities to evaluate all trigonometric functions. However, you may need to use multiple identities to express the function in terms of the given function.

Q: Are the Pythagorean identities only useful for evaluating trigonometric functions?

A: No, the Pythagorean identities are also useful for deriving other trigonometric identities and for solving trigonometric equations.

Q: Can I use the Pythagorean identities to evaluate trigonometric functions in terms of other trigonometric functions?

A: Yes, you can use the Pythagorean identities to express one trigonometric function in terms of another. For example, you can use the identity $\cot \theta = \frac{1}{\tan \theta}$ to express the cotangent of an angle in terms of the tangent of the angle.

Q: Are the Pythagorean identities only useful for right triangles?

A: No, the Pythagorean identities are useful for all triangles, not just right triangles.

Q: Can I use the Pythagorean identities to evaluate trigonometric functions in terms of the sine and cosine of an angle?

A: Yes, you can use the Pythagorean identities to express one trigonometric function in terms of the sine and cosine of an angle. For example, you can use the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$ to express the tangent of an angle in terms of the sine and cosine of the angle.

Q: Are the Pythagorean identities only useful for evaluating trigonometric functions in terms of the sine and cosine of an angle?

A: No, the Pythagorean identities are useful for evaluating trigonometric functions in terms of any trigonometric function.

Q: Can I use the Pythagorean identities to evaluate trigonometric functions in terms of the secant and cosecant of an angle?

A: Yes, you can use the Pythagorean identities to express one trigonometric function in terms of the secant and cosecant of an angle. For example, you can use the identity $\sec \theta = \frac{1}{\cos \theta}$ to express the secant of an angle in terms of the cosine of the angle.

Q: Are the Pythagorean identities only useful for evaluating trigonometric functions in terms of the secant and cosecant of an angle?

A: No, the Pythagorean identities are useful for evaluating trigonometric functions in terms of any trigonometric function.

Q: Can I use the Pythagorean identities to evaluate trigonometric functions in terms of the tangent and cotangent of an angle?

A: Yes, you can use the Pythagorean identities to express one trigonometric function in terms of the tangent and cotangent of an angle. For example, you can use the identity $\cot \theta = \frac{1}{\tan \theta}$ to express the cotangent of an angle in terms of the tangent of the angle.

Q: Are the Pythagorean identities only useful for evaluating trigonometric functions in terms of the tangent and cotangent of an angle?

A: No, the Pythagorean identities are useful for evaluating trigonometric functions in terms of any trigonometric function.

Conclusion

In this article, we have answered some frequently asked questions about evaluating trigonometric functions using Pythagorean identities. We have shown that the Pythagorean identities are a powerful tool for evaluating trigonometric functions and can be used to express one trigonometric function in terms of another.

References

• [1] "Trigonometry" by Michael Corral, 2015.
• [2] "Calculus" by Michael Spivak, 2008.

Further Reading

• For more information on the Pythagorean identities, see [1].
• For more information on trigonometry, see [2].

Code

import math

def evaluate_tan(theta):
    sec_theta = 8/5
    tan_squared_theta = (sec_theta**2) - 1
    tan_theta = -math.sqrt(tan_squared_theta)
    return tan_theta

def evaluate_cot(theta):
    tan_theta = evaluate_tan(theta)
    cot_theta = 1 / tan_theta
    return cot_theta

theta = math.acos(-3/5)
tan_theta = evaluate_tan(theta)
cot_theta = evaluate_cot(theta)

print("The value of tan(θ) is: ", tan_theta)
print("The value of cot(θ) is: ", cot_theta)

Note: The code above is for demonstration purposes only and is not intended to be used for actual calculations.