Determine The Value Of Sin ⁡ ( A + B \sin (A+B Sin ( A + B ] If 5 Cos ⁡ A − 4 = 0 5 \cos A - 4 = 0 5 Cos A − 4 = 0 And 12 Tan ⁡ B + 5 = 0 12 \tan B + 5 = 0 12 Tan B + 5 = 0 ; A A A And $B \in [0^{\circ}, 180^{\circ}].

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Introduction

In this article, we will explore the problem of determining the value of $\sin (A+B)$ given two equations involving trigonometric functions. We will use trigonometric identities to solve for the values of $A$ and $B$, and then find the value of $\sin (A+B)$. This problem requires a deep understanding of trigonometric functions and their relationships.

Given Equations

We are given two equations:

1. $5 \cos A - 4 = 0$
2. $12 \tan B + 5 = 0$

These equations involve the trigonometric functions cosine and tangent, respectively. We will use these equations to solve for the values of $A$ and $B$.

Solving for $A$

To solve for $A$, we can start by isolating $\cos A$ in the first equation:

$$5 \cos A - 4 = 0$$

$$\cos A = \frac{4}{5}$$

Since $A \in [0^{\circ}, 180^{\circ}]$, we can use the inverse cosine function to find the value of $A$:

$$A = \cos^{-1} \left( \frac{4}{5} \right)$$

Using a calculator, we find that $A \approx 36.87^{\circ}$.

Solving for $B$

To solve for $B$, we can start by isolating $\tan B$ in the second equation:

$$12 \tan B + 5 = 0$$

$$\tan B = -\frac{5}{12}$$

Since $B \in [0^{\circ}, 180^{\circ}]$, we can use the inverse tangent function to find the value of $B$:

$$B = \tan^{-1} \left( -\frac{5}{12} \right)$$

Using a calculator, we find that $B \approx 169.65^{\circ}$.

Finding the Value of $\sin (A+B)$

Now that we have found the values of $A$ and $B$, we can use the angle addition formula for sine to find the value of $\sin (A+B)$:

$$\sin (A+B) = \sin A \cos B + \cos A \sin B$$

We can substitute the values of $\sin A$, $\cos A$, $\sin B$, and $\cos B$ into this formula:

$$\sin (A+B) = \sin 36.87^{\circ} \cos 169.65^{\circ} + \cos 36.87^{\circ} \sin 169.65^{\circ}$$

Using a calculator, we find that:

$$\sin (A+B) \approx -0.75$$

Conclusion

In this article, we used trigonometric identities to solve for the values of $A$ and $B$ given two equations involving trigonometric functions. We then used the angle addition formula for sine to find the value of $\sin (A+B)$. This problem required a deep understanding of trigonometric functions and their relationships.

Trigonometric Identities Used

• $\cos^{-1} x$ (inverse cosine function)
• $\tan^{-1} x$ (inverse tangent function)
• $\sin (A+B) = \sin A \cos B + \cos A \sin B$ (angle addition formula for sine)

Future Work

This problem can be extended to find the values of $\cos (A+B)$ and $\tan (A+B)$ using the angle addition formulas for cosine and tangent, respectively.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak

Appendix

The following is a list of trigonometric identities that were used in this article:

• $\sin^2 x + \cos^2 x = 1$
• $\tan x = \frac{\sin x}{\cos x}$
• $\cot x = \frac{\cos x}{\sin x}$

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Introduction

In our previous article, we explored the problem of determining the value of $\sin (A+B)$ given two equations involving trigonometric functions. We used trigonometric identities to solve for the values of $A$ and $B$, and then found the value of $\sin (A+B)$. In this article, we will answer some common questions related to this problem.

Q: What are the given equations?

A: The given equations are:

1. $5 \cos A - 4 = 0$
2. $12 \tan B + 5 = 0$

Q: How do we solve for $A$?

A: To solve for $A$, we can start by isolating $\cos A$ in the first equation:

$$5 \cos A - 4 = 0$$

$$\cos A = \frac{4}{5}$$

Since $A \in [0^{\circ}, 180^{\circ}]$, we can use the inverse cosine function to find the value of $A$:

$$A = \cos^{-1} \left( \frac{4}{5} \right)$$

Q: How do we solve for $B$?

A: To solve for $B$, we can start by isolating $\tan B$ in the second equation:

$$12 \tan B + 5 = 0$$

$$\tan B = -\frac{5}{12}$$

Since $B \in [0^{\circ}, 180^{\circ}]$, we can use the inverse tangent function to find the value of $B$:

$$B = \tan^{-1} \left( -\frac{5}{12} \right)$$

Q: What is the value of $\sin (A+B)$?

A: We can use the angle addition formula for sine to find the value of $\sin (A+B)$:

$$\sin (A+B) = \sin A \cos B + \cos A \sin B$$

We can substitute the values of $\sin A$, $\cos A$, $\sin B$, and $\cos B$ into this formula:

$$\sin (A+B) = \sin 36.87^{\circ} \cos 169.65^{\circ} + \cos 36.87^{\circ} \sin 169.65^{\circ}$$

Using a calculator, we find that:

$$\sin (A+B) \approx -0.75$$

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

• Not isolating $\cos A$ and $\tan B$ in the given equations
• Not using the inverse cosine and inverse tangent functions to find the values of $A$ and $B$
• Not substituting the values of $\sin A$, $\cos A$, $\sin B$, and $\cos B$ into the angle addition formula for sine

Q: How can I apply this problem to real-world situations?

A: This problem can be applied to real-world situations such as:

• Finding the value of the sine of an angle in a right triangle
• Determining the value of the sine of an angle in a circular motion
• Calculating the value of the sine of an angle in a trigonometric function

Conclusion

In this article, we answered some common questions related to the problem of determining the value of $\sin (A+B)$ given two equations involving trigonometric functions. We hope that this article has provided you with a better understanding of this problem and how to apply it to real-world situations.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak

Appendix

The following is a list of trigonometric identities that were used in this article:

• $\sin^2 x + \cos^2 x = 1$
• $\tan x = \frac{\sin x}{\cos x}$
• $\cot x = \frac{\cos x}{\sin x}$

These identities can be used to derive the angle addition formulas for sine, cosine, and tangent.