Expressing Sine Functions in Terms of p

Oct 13, 2025 by miodeya.com 149 views

=====================================================

Introduction

In this article, we will explore the concept of expressing trigonometric functions in terms of a given value, in this case, $p = \sin 40^{\circ}$ . We will use the given value to write the following expressions in terms of $p$ : $\sin 140^{\circ}$ , $\sin 220^{\circ}$ , $\sin (-140^{\circ})$ , $\cos 50^{\circ}$ , $\sin 50^{\circ}$ , and $\tan 50^{\circ}$ .

Expressing Sine Functions in Terms of p

(a) $\sin 140^{\circ}$

We know that $\sin 140^{\circ} = \sin (180^{\circ} - 40^{\circ})$ . Using the identity $\sin (180^{\circ} - x) = \sin x$ , we can rewrite $\sin 140^{\circ}$ as $\sin 40^{\circ} = p$ .

(b) $\sin 220^{\circ}$

We know that $\sin 220^{\circ} = \sin (180^{\circ} + 40^{\circ})$ . Using the identity $\sin (180^{\circ} + x) = -\sin x$ , we can rewrite $\sin 220^{\circ}$ as $-\sin 40^{\circ} = -p$ .

(c) $\sin (-140^{\circ})$

We know that $\sin (-140^{\circ}) = \sin (180^{\circ} + 40^{\circ})$ . Using the identity $\sin (180^{\circ} + x) = -\sin x$ , we can rewrite $\sin (-140^{\circ})$ as $-\sin 40^{\circ} = -p$ .

Expressing Cosine and Tangent Functions in Terms of p

(d) $\cos 50^{\circ}$

We know that $\cos 50^{\circ} = \cos (90^{\circ} - 40^{\circ})$ . Using the identity $\cos (90^{\circ} - x) = \sin x$ , we can rewrite $\cos 50^{\circ}$ as $\sin 40^{\circ} = p$ .

(e) $\sin 50^{\circ}$

We know that $\sin 50^{\circ} = \sin (90^{\circ} - 40^{\circ})$ . Using the identity $\sin (90^{\circ} - x) = \cos x$ , we can rewrite $\sin 50^{\circ}$ as $\cos 40^{\circ}$ . However, we can also express $\sin 50^{\circ}$ in terms of $p$ using the identity $\sin x = \cos (90^{\circ} - x)$ . Therefore, $\sin 50^{\circ} = \cos (90^{\circ} - 50^{\circ}) = \cos 40^{\circ} = \sqrt{1 - \sin^2 40^{\circ}} = \sqrt{1 - p^2}$ .

(f) $\tan 50^{\circ}$

We know that $\tan 50^{\circ} = \tan (90^{\circ} - 40^{\circ})$ . Using the identity $\tan (90^{\circ} - x) = \cot x$ , we can rewrite $\tan 50^{\circ}$ as $\cot 40^{\circ}$ . However, we can also express $\tan 50^{\circ}$ in terms of $p$ using the identity $\tan x = \frac{\sin x}{\cos x}$ . Therefore, $\tan 50^{\circ} = \frac{\sin 50^{\circ}}{\cos 50^{\circ}} = \frac{\sqrt{1 - \sin^2 40^{\circ}}}{\sin 40^{\circ}} = \frac{\sqrt{1 - p^2}}{p}$ .

Conclusion

In this article, we have expressed various trigonometric functions in terms of a given value, $p = \sin 40^{\circ}$ . We have used various trigonometric identities to rewrite the expressions in terms of $p$ . The resulting expressions are:

$\sin 140^{\circ} = p$
$\sin 220^{\circ} = -p$
$\sin (-140^{\circ}) = -p$
$\cos 50^{\circ} = p$
$\sin 50^{\circ} = \sqrt{1 - p^2}$
$\tan 50^{\circ} = \frac{\sqrt{1 - p^2}}{p}$

These expressions demonstrate the power of trigonometric identities in simplifying and manipulating trigonometric functions.

=====================================================

Introduction

In our previous article, we explored the concept of expressing trigonometric functions in terms of a given value, in this case, $p = \sin 40^{\circ}$ . We used various trigonometric identities to rewrite the expressions in terms of $p$ . In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the value of $\sin 140^{\circ}$ in terms of $p$ ?

A: The value of $\sin 140^{\circ}$ in terms of $p$ is $p$ .

Q: What is the value of $\sin 220^{\circ}$ in terms of $p$ ?

A: The value of $\sin 220^{\circ}$ in terms of $p$ is $-p$ .

Q: What is the value of $\sin (-140^{\circ})$ in terms of $p$ ?

A: The value of $\sin (-140^{\circ})$ in terms of $p$ is $-p$ .

Q: What is the value of $\cos 50^{\circ}$ in terms of $p$ ?

A: The value of $\cos 50^{\circ}$ in terms of $p$ is $p$ .

Q: What is the value of $\sin 50^{\circ}$ in terms of $p$ ?

A: The value of $\sin 50^{\circ}$ in terms of $p$ is $\sqrt{1 - p^2}$ .

Q: What is the value of $\tan 50^{\circ}$ in terms of $p$ ?

A: The value of $\tan 50^{\circ}$ in terms of $p$ is $\frac{\sqrt{1 - p^2}}{p}$ .

Q: How do I use trigonometric identities to simplify trigonometric expressions?

A: To simplify trigonometric expressions, you can use various trigonometric identities such as the sum and difference formulas, the double-angle formulas, and the half-angle formulas. You can also use the Pythagorean identity to rewrite expressions in terms of sine and cosine.

Q: What are some common trigonometric identities that I should know?

A: Some common trigonometric identities that you should know include:

$\sin (A + B) = \sin A \cos B + \cos A \sin B$
$\sin (A - B) = \sin A \cos B - \cos A \sin B$
$\cos (A + B) = \cos A \cos B - \sin A \sin B$
$\cos (A - B) = \cos A \cos B + \sin A \sin B$
$\tan A = \frac{\sin A}{\cos A}$
$\cot A = \frac{\cos A}{\sin A}$
$\csc A = \frac{1}{\sin A}$
$\sec A = \frac{1}{\cos A}$

Conclusion

In this article, we have answered some frequently asked questions related to expressing trigonometric functions in terms of a given value. We have also provided some common trigonometric identities that you should know. By using these identities, you can simplify and manipulate trigonometric expressions with ease.

Additional Resources

If you want to learn more about trigonometric identities and how to use them to simplify trigonometric expressions, we recommend checking out the following resources:

Khan Academy: Trigonometry
Mathway: Trigonometric Identities
Wolfram Alpha: Trigonometric Identities

These resources provide a comprehensive overview of trigonometric identities and how to use them to simplify trigonometric expressions.