Find $\sin \theta$ And $\tan \theta$ If $\cos \theta = -\frac{3}{4}$ And $\sin \theta \ \textgreater \ 0$.

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Introduction

In trigonometry, the sine, cosine, and tangent functions are used to describe the relationships between the angles and side lengths of triangles. Given the value of one of these functions, we can use the Pythagorean identity to find the values of the other two functions. In this article, we will discuss how to find sin⁡θ\sin \theta and tan⁡θ\tan \theta if cos⁡θ=−34\cos \theta = -\frac{3}{4} and sin⁡θ \textgreater 0\sin \theta \ \textgreater \ 0.

Understanding the Pythagorean Identity

The Pythagorean identity states that for any angle θ\theta, the following equation holds:

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

This identity can be used to find the values of sin⁡θ\sin \theta and tan⁡θ\tan \theta if the value of cos⁡θ\cos \theta is known.

Finding sin⁡θ\sin \theta

Since we are given that cos⁡θ=−34\cos \theta = -\frac{3}{4} and sin⁡θ \textgreater 0\sin \theta \ \textgreater \ 0, we can use the Pythagorean identity to find the value of sin⁡θ\sin \theta. We can start by rearranging the Pythagorean identity to isolate sin⁡2θ\sin^2 \theta:

sin⁡2θ=1−cos⁡2θ\sin^2 \theta = 1 - \cos^2 \theta

Substituting the value of cos⁡θ\cos \theta, we get:

sin⁡2θ=1−(−34)2\sin^2 \theta = 1 - \left(-\frac{3}{4}\right)^2

Simplifying the expression, we get:

sin⁡2θ=1−916\sin^2 \theta = 1 - \frac{9}{16}

sin⁡2θ=716\sin^2 \theta = \frac{7}{16}

Since sin⁡θ \textgreater 0\sin \theta \ \textgreater \ 0, we can take the square root of both sides to find the value of sin⁡θ\sin \theta:

sin⁡θ=716\sin \theta = \sqrt{\frac{7}{16}}

sin⁡θ=74\sin \theta = \frac{\sqrt{7}}{4}

Finding tan⁡θ\tan \theta

Now that we have found the value of sin⁡θ\sin \theta, we can use the definition of the tangent function to find the value of tan⁡θ\tan \theta:

tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

Substituting the values of sin⁡θ\sin \theta and cos⁡θ\cos \theta, we get:

tan⁡θ=74−34\tan \theta = \frac{\frac{\sqrt{7}}{4}}{-\frac{3}{4}}

Simplifying the expression, we get:

tan⁡θ=−73\tan \theta = -\frac{\sqrt{7}}{3}

Conclusion

In this article, we discussed how to find sin⁡θ\sin \theta and tan⁡θ\tan \theta if cos⁡θ=−34\cos \theta = -\frac{3}{4} and sin⁡θ \textgreater 0\sin \theta \ \textgreater \ 0. We used the Pythagorean identity to find the value of sin⁡θ\sin \theta, and then used the definition of the tangent function to find the value of tan⁡θ\tan \theta. The final values of sin⁡θ\sin \theta and tan⁡θ\tan \theta are 74\frac{\sqrt{7}}{4} and −73-\frac{\sqrt{7}}{3}, respectively.

Example Use Case

The values of sin⁡θ\sin \theta and tan⁡θ\tan \theta can be used in a variety of applications, such as:

  • Calculating the height of a building or a mountain using the angle of elevation
  • Determining the distance to a object using the angle of depression
  • Finding the area of a triangle using the sine and cosine functions

Step-by-Step Solution

Here is a step-by-step solution to the problem:

  1. Start with the Pythagorean identity: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  2. Rearrange the identity to isolate sin⁡2θ\sin^2 \theta: sin⁡2θ=1−cos⁡2θ\sin^2 \theta = 1 - \cos^2 \theta
  3. Substitute the value of cos⁡θ\cos \theta: sin⁡2θ=1−(−34)2\sin^2 \theta = 1 - \left(-\frac{3}{4}\right)^2
  4. Simplify the expression: sin⁡2θ=1−916\sin^2 \theta = 1 - \frac{9}{16}
  5. Simplify further: sin⁡2θ=716\sin^2 \theta = \frac{7}{16}
  6. Take the square root of both sides: sin⁡θ=716\sin \theta = \sqrt{\frac{7}{16}}
  7. Simplify the expression: sin⁡θ=74\sin \theta = \frac{\sqrt{7}}{4}
  8. Use the definition of the tangent function: tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
  9. Substitute the values of sin⁡θ\sin \theta and cos⁡θ\cos \theta: tan⁡θ=74−34\tan \theta = \frac{\frac{\sqrt{7}}{4}}{-\frac{3}{4}}
  10. Simplify the expression: tan⁡θ=−73\tan \theta = -\frac{\sqrt{7}}{3}

Frequently Asked Questions

  • Q: What is the Pythagorean identity? A: The Pythagorean identity is a fundamental equation in trigonometry that states sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.
  • Q: How do I find the value of sin⁡θ\sin \theta if cos⁡θ\cos \theta is known? A: You can use the Pythagorean identity to find the value of sin⁡θ\sin \theta by rearranging the identity to isolate sin⁡2θ\sin^2 \theta and then taking the square root of both sides.
  • Q: What is the definition of the tangent function? A: The tangent function is defined as tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.

Final Answer

The final values of sin⁡θ\sin \theta and tan⁡θ\tan \theta are 74\frac{\sqrt{7}}{4} and −73-\frac{\sqrt{7}}{3}, respectively.

Introduction

In our previous article, we discussed how to find sin⁡θ\sin \theta and tan⁡θ\tan \theta if cos⁡θ=−34\cos \theta = -\frac{3}{4} and sin⁡θ \textgreater 0\sin \theta \ \textgreater \ 0. We used the Pythagorean identity to find the value of sin⁡θ\sin \theta, and then used the definition of the tangent function to find the value of tan⁡θ\tan \theta. In this article, we will answer some frequently asked questions about finding sin⁡θ\sin \theta and tan⁡θ\tan \theta.

Q&A

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental equation in trigonometry that states sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1. This identity can be used to find the values of sin⁡θ\sin \theta and tan⁡θ\tan \theta if the value of cos⁡θ\cos \theta is known.

Q: How do I find the value of sin⁡θ\sin \theta if cos⁡θ\cos \theta is known?

A: You can use the Pythagorean identity to find the value of sin⁡θ\sin \theta by rearranging the identity to isolate sin⁡2θ\sin^2 \theta and then taking the square root of both sides. For example, if cos⁡θ=−34\cos \theta = -\frac{3}{4}, you can substitute this value into the Pythagorean identity to find the value of sin⁡θ\sin \theta.

Q: What is the definition of the tangent function?

A: The tangent function is defined as tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}. This function can be used to find the value of tan⁡θ\tan \theta if the values of sin⁡θ\sin \theta and cos⁡θ\cos \theta are known.

Q: How do I find the value of tan⁡θ\tan \theta if sin⁡θ\sin \theta and cos⁡θ\cos \theta are known?

A: You can use the definition of the tangent function to find the value of tan⁡θ\tan \theta by substituting the values of sin⁡θ\sin \theta and cos⁡θ\cos \theta into the definition. For example, if sin⁡θ=74\sin \theta = \frac{\sqrt{7}}{4} and cos⁡θ=−34\cos \theta = -\frac{3}{4}, you can substitute these values into the definition of the tangent function to find the value of tan⁡θ\tan \theta.

Q: What if sin⁡θ\sin \theta is not positive?

A: If sin⁡θ\sin \theta is not positive, you will need to use the negative square root to find the value of sin⁡θ\sin \theta. For example, if sin⁡θ=−74\sin \theta = -\frac{\sqrt{7}}{4}, you can use the negative square root to find the value of sin⁡θ\sin \theta.

Q: Can I use the Pythagorean identity to find the value of tan⁡θ\tan \theta?

A: No, the Pythagorean identity can only be used to find the values of sin⁡θ\sin \theta and cos⁡θ\cos \theta. To find the value of tan⁡θ\tan \theta, you will need to use the definition of the tangent function.

Q: What if I don't know the value of cos⁡θ\cos \theta?

A: If you don't know the value of cos⁡θ\cos \theta, you will not be able to use the Pythagorean identity to find the value of sin⁡θ\sin \theta. In this case, you will need to use other methods to find the value of sin⁡θ\sin \theta.

Example Use Cases

The values of sin⁡θ\sin \theta and tan⁡θ\tan \theta can be used in a variety of applications, such as:

  • Calculating the height of a building or a mountain using the angle of elevation
  • Determining the distance to a object using the angle of depression
  • Finding the area of a triangle using the sine and cosine functions

Step-by-Step Solution

Here is a step-by-step solution to the problem:

  1. Start with the Pythagorean identity: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  2. Rearrange the identity to isolate sin⁡2θ\sin^2 \theta: sin⁡2θ=1−cos⁡2θ\sin^2 \theta = 1 - \cos^2 \theta
  3. Substitute the value of cos⁡θ\cos \theta: sin⁡2θ=1−(−34)2\sin^2 \theta = 1 - \left(-\frac{3}{4}\right)^2
  4. Simplify the expression: sin⁡2θ=1−916\sin^2 \theta = 1 - \frac{9}{16}
  5. Simplify further: sin⁡2θ=716\sin^2 \theta = \frac{7}{16}
  6. Take the square root of both sides: sin⁡θ=716\sin \theta = \sqrt{\frac{7}{16}}
  7. Simplify the expression: sin⁡θ=74\sin \theta = \frac{\sqrt{7}}{4}
  8. Use the definition of the tangent function: tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
  9. Substitute the values of sin⁡θ\sin \theta and cos⁡θ\cos \theta: tan⁡θ=74−34\tan \theta = \frac{\frac{\sqrt{7}}{4}}{-\frac{3}{4}}
  10. Simplify the expression: tan⁡θ=−73\tan \theta = -\frac{\sqrt{7}}{3}

Frequently Asked Questions

  • Q: What is the Pythagorean identity? A: The Pythagorean identity is a fundamental equation in trigonometry that states sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.
  • Q: How do I find the value of sin⁡θ\sin \theta if cos⁡θ\cos \theta is known? A: You can use the Pythagorean identity to find the value of sin⁡θ\sin \theta by rearranging the identity to isolate sin⁡2θ\sin^2 \theta and then taking the square root of both sides.
  • Q: What is the definition of the tangent function? A: The tangent function is defined as tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.

Final Answer

The final values of sin⁡θ\sin \theta and tan⁡θ\tan \theta are 74\frac{\sqrt{7}}{4} and −73-\frac{\sqrt{7}}{3}, respectively.