In A Right Triangle, The Acute Angles Have The Relationship $\sin \left(x+12^{\circ}\right)=\cos \left(18^{\circ}+2x\right$\]. What Is The Value Of $x$? What Is The Measure Of The Smaller Angle?Use 1-2 Sentences To Explain How You

Introduction

In this article, we will explore the relationship between the acute angles in a right triangle and how to solve trigonometric equations involving sine and cosine functions. We will use the given equation $\sin \left(x+12^{\circ}\right)=\cos \left(18^{\circ}+2x\right)$ to find the value of $x$ and the measure of the smaller angle.

Understanding the Trigonometric Equation

The given equation involves the sine and cosine functions of two different angles. To solve this equation, we need to use the trigonometric identity that relates the sine and cosine functions: $\sin \left(x\right)=\cos \left(90^{\circ}-x\right)$. We can rewrite the given equation as:

$$\sin \left(x+12^{\circ}\right)=\cos \left(18^{\circ}+2x\right)$$

$$\sin \left(x+12^{\circ}\right)=\sin \left(90^{\circ}-\left(18^{\circ}+2x\right)\right)$$

Using Trigonometric Identities

We can use the trigonometric identity $\sin \left(x\right)=\sin \left(180^{\circ}-x\right)$ to rewrite the equation as:

$$\sin \left(x+12^{\circ}\right)=\sin \left(180^{\circ}-\left(18^{\circ}+2x\right)\right)$$

$$\sin \left(x+12^{\circ}\right)=\sin \left(162^{\circ}-2x\right)$$

Equating Angles

Since the sine function is periodic with a period of $180^{\circ}$, we can equate the angles inside the sine functions:

$$x+12^{\circ}=162^{\circ}-2x$$

Solving for $x$

We can now solve for $x$ by isolating the variable:

$$x+12^{\circ}=162^{\circ}-2x$$

$$3x=150^{\circ}$$

$$x=50^{\circ}$$

Finding the Measure of the Smaller Angle

Since the sum of the acute angles in a right triangle is $90^{\circ}$, we can find the measure of the smaller angle by subtracting the value of $x$ from $90^{\circ}$:

$$90^{\circ}-50^{\circ}=40^{\circ}$$

Conclusion

In this article, we used the given trigonometric equation to find the value of $x$ and the measure of the smaller angle in a right triangle. We used the trigonometric identity that relates the sine and cosine functions and the periodicity of the sine function to solve the equation. The value of $x$ is $50^{\circ}$, and the measure of the smaller angle is $40^{\circ}$.

Additional Tips and Tricks

• When solving trigonometric equations, it's essential to use the correct trigonometric identities and to consider the periodicity of the trigonometric functions.
• When equating angles inside the sine or cosine functions, make sure to consider the periodicity of the functions and to use the correct trigonometric identities.
• When solving for the value of $x$, make sure to isolate the variable and to use the correct algebraic operations.

Common Mistakes to Avoid

• Not using the correct trigonometric identities when solving the equation.
• Not considering the periodicity of the trigonometric functions when equating angles.
• Not isolating the variable when solving for the value of $x$.

Real-World Applications

Trigonometric equations have numerous real-world applications in various fields, including:

• Physics: Trigonometric equations are used to describe the motion of objects in terms of their position, velocity, and acceleration.
• Engineering: Trigonometric equations are used to design and analyze the performance of mechanical systems, such as gears and linkages.
• Computer Science: Trigonometric equations are used in computer graphics to create 3D models and to simulate the motion of objects.

Conclusion

$$$$$$$$

Q: What is the relationship between the sine and cosine functions?

A: The sine and cosine functions are related by the trigonometric identity $\sin \left(x\right)=\cos \left(90^{\circ}-x\right)$. This identity allows us to rewrite the given equation in terms of the cosine function.

Q: How do I use the periodicity of the sine function to solve the equation?

A: The sine function is periodic with a period of $180^{\circ}$. This means that the sine function repeats itself every $180^{\circ}$. We can use this property to rewrite the equation and solve for the value of $x$.

Q: What is the significance of the angle $90^{\circ}$ in the trigonometric identity?

A: The angle $90^{\circ}$ is a special angle in trigonometry that represents a right angle. The trigonometric identity $\sin \left(x\right)=\cos \left(90^{\circ}-x\right)$ shows that the sine and cosine functions are related by a $90^{\circ}$ rotation.

Q: How do I find the measure of the smaller angle in a right triangle?

A: To find the measure of the smaller angle in a right triangle, we can subtract the value of $x$ from $90^{\circ}$. This will give us the measure of the smaller angle.

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

• Not using the correct trigonometric identities
• Not considering the periodicity of the trigonometric functions
• Not isolating the variable when solving for the value of $x$

Q: What are some real-world applications of trigonometric equations?

A: Trigonometric equations have numerous real-world applications in various fields, including:

• Physics: Trigonometric equations are used to describe the motion of objects in terms of their position, velocity, and acceleration.
• Engineering: Trigonometric equations are used to design and analyze the performance of mechanical systems, such as gears and linkages.
• Computer Science: Trigonometric equations are used in computer graphics to create 3D models and to simulate the motion of objects.

Q: How do I use the trigonometric identity to rewrite the equation?

A: To use the trigonometric identity to rewrite the equation, we can substitute the expression $\cos \left(90^{\circ}-x\right)$ for $\sin \left(x\right)$ in the given equation.

Q: What is the value of $x$ in the given equation?

A: The value of $x$ in the given equation is $50^{\circ}$.

Q: What is the measure of the smaller angle in the right triangle?

A: The measure of the smaller angle in the right triangle is $40^{\circ}$.

Q: How do I check my answer to make sure it is correct?

A: To check your answer, you can plug the value of $x$ back into the original equation and make sure that it is true. You can also use a calculator to check your answer.

Q: What are some tips for solving trigonometric equations?

A: Some tips for solving trigonometric equations include:

• Using the correct trigonometric identities
• Considering the periodicity of the trigonometric functions
• Isolating the variable when solving for the value of $x$
• Checking your answer to make sure it is correct