Cosine Function with Negative Angle

Oct 13, 2025 by windailys.com 118 views

Introduction

In this article, we will simplify the given trigonometric equation $\frac{\sin (180-\theta) \cdot \cos (90-\theta)}{\cos (-\theta)}=1$ . This equation involves various trigonometric functions, including sine, cosine, and their properties. We will use the fundamental identities and properties of trigonometric functions to simplify the equation and find the solution.

Understanding the Trigonometric Functions

Before we proceed with simplifying the equation, let's understand the trigonometric functions involved.

Sine Function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. It is denoted by $\sin \theta$ .
Cosine Function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. It is denoted by $\cos \theta$ .
Cosine Function with Negative Angle: The cosine function with a negative angle is defined as $\cos (-\theta) = \cos \theta$ .

Simplifying the Equation

Now, let's simplify the given equation using the properties of trigonometric functions.

$\frac{\sin (180-\theta) \cdot \cos (90-\theta)}{\cos (-\theta)}=1$

Using the property of sine function, we can rewrite $\sin (180-\theta)$ as $\sin \theta$ .

$\frac{\sin \theta \cdot \cos (90-\theta)}{\cos (-\theta)}=1$

Using the property of cosine function, we can rewrite $\cos (90-\theta)$ as $\sin \theta$ .

$\frac{\sin \theta \cdot \sin \theta}{\cos (-\theta)}=1$

Using the property of cosine function with negative angle, we can rewrite $\cos (-\theta)$ as $\cos \theta$ .

$\frac{\sin^2 \theta}{\cos \theta}=1$

Using the Pythagorean Identity

We can use the Pythagorean identity to simplify the equation further.

$\sin^2 \theta + \cos^2 \theta = 1$

Rearranging the terms, we get:

$\sin^2 \theta = 1 - \cos^2 \theta$

Substituting this expression into the equation, we get:

$\frac{1 - \cos^2 \theta}{\cos \theta}=1$

Simplifying the Equation Further

We can simplify the equation further by multiplying both sides by $\cos \theta$ .

$1 - \cos^2 \theta = \cos \theta$

Using the Quadratic Formula

We can use the quadratic formula to solve the equation.

$\cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$ , $b = -1$ , and $c = 0$ .

$\cos \theta = \frac{1 \pm \sqrt{1 - 4(1)(0)}}{2(1)}$

$\cos \theta = \frac{1 \pm \sqrt{1}}{2}$

$\cos \theta = \frac{1 \pm 1}{2}$

Solving for $\cos \theta$

We have two possible solutions for $\cos \theta$ .

$\cos \theta = \frac{1 + 1}{2}$

$\cos \theta = 1$

$\cos \theta = \frac{1 - 1}{2}$

$\cos \theta = 0$

Conclusion

In this article, we simplified the given trigonometric equation $\frac{\sin (180-\theta) \cdot \cos (90-\theta)}{\cos (-\theta)}=1$ using the properties of trigonometric functions. We used the Pythagorean identity to simplify the equation further and solved for $\cos \theta$ using the quadratic formula. The solutions for $\cos \theta$ are $\cos \theta = 1$ and $\cos \theta = 0$ .

Final Answer

The final answer is $\boxed{1}$ and $\boxed{0}$ .

Introduction

In our previous article, we simplified the trigonometric equation $\frac{\sin (180-\theta) \cdot \cos (90-\theta)}{\cos (-\theta)}=1$ . In this article, we will answer some frequently asked questions related to simplifying trigonometric equations.

Q1: What are the common trigonometric identities used to simplify equations?

A1: The common trigonometric identities used to simplify equations are:

Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$
Quadratic Formula: $\cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Sine and Cosine Functions: $\sin \theta = \frac{opposite}{hypotenuse}$ and $\cos \theta = \frac{adjacent}{hypotenuse}$

Q2: How do I simplify a trigonometric equation with multiple terms?

A2: To simplify a trigonometric equation with multiple terms, follow these steps:

Identify the common terms: Identify the common terms in the equation and group them together.
Use trigonometric identities: Use trigonometric identities to simplify the equation.
Combine like terms: Combine like terms to simplify the equation further.

Q3: What is the difference between $\sin \theta$ and $\cos \theta$ ?

A3: The difference between $\sin \theta$ and $\cos \theta$ is:

Sine Function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
Cosine Function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.

Q4: How do I solve a trigonometric equation with a negative angle?

A4: To solve a trigonometric equation with a negative angle, follow these steps:

Use the property of cosine function with negative angle: $\cos (-\theta) = \cos \theta$
Simplify the equation: Simplify the equation using trigonometric identities.
Solve for $\theta$ : Solve for $\theta$ using algebraic methods.

Q5: What are some common mistakes to avoid when simplifying trigonometric equations?

A5: Some common mistakes to avoid when simplifying trigonometric equations are:

Not using trigonometric identities: Not using trigonometric identities can lead to complex and difficult-to-solve equations.
Not combining like terms: Not combining like terms can lead to unnecessary complexity in the equation.
Not checking for extraneous solutions: Not checking for extraneous solutions can lead to incorrect solutions.

Conclusion

In this article, we answered some frequently asked questions related to simplifying trigonometric equations. We covered topics such as common trigonometric identities, simplifying equations with multiple terms, and solving equations with negative angles. We also discussed some common mistakes to avoid when simplifying trigonometric equations.

Final Answer

The final answer is that simplifying trigonometric equations requires a deep understanding of trigonometric identities and algebraic methods. By following the steps outlined in this article, you can simplify even the most complex trigonometric equations.

Additional Resources

For more information on simplifying trigonometric equations, check out the following resources:

Trigonometric Identities: A comprehensive list of trigonometric identities.
Algebraic Methods: A guide to algebraic methods for solving trigonometric equations.
Trigonometric Equations: A collection of trigonometric equations with solutions.

For more information on trigonometric equations, check out the following related articles:

Simplifying Trigonometric Equations: A step-by-step guide to simplifying trigonometric equations.
Solving Trigonometric Equations: A guide to solving trigonometric equations using algebraic methods.
Trigonometric Equations with Negative Angles: A guide to solving trigonometric equations with negative angles.