Given: $\tan \theta=\frac{\sqrt{15}}{3}$ And $\cos \theta=\frac{\sqrt{10}}{4}$Find: $\sin \theta$

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Introduction

In trigonometry, the sine, cosine, and tangent of an angle in a right-angled triangle are defined as the ratios of the lengths of the sides of the triangle. Given the tangent and cosine of an angle, we can use these definitions to find the sine of the angle. In this article, we will use the given values of tan⁡θ=153\tan \theta=\frac{\sqrt{15}}{3} and cos⁡θ=104\cos \theta=\frac{\sqrt{10}}{4} to find the value of sin⁡θ\sin \theta.

Recalling Trigonometric Identities

Before we proceed, let's recall some important trigonometric identities that we will use to find the sine of the angle.

  • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

Using the Pythagorean Identity

We can use the Pythagorean identity to find the value of sin⁡θ\sin \theta. Since sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, we can rearrange this equation to get sin⁡2θ=1−cos⁡2θ\sin^2 \theta = 1 - \cos^2 \theta. Substituting the given value of cos⁡θ=104\cos \theta=\frac{\sqrt{10}}{4}, we get:

sin⁡2θ=1−(104)2\sin^2 \theta = 1 - \left(\frac{\sqrt{10}}{4}\right)^2

Simplifying this equation, we get:

sin⁡2θ=1−1016\sin^2 \theta = 1 - \frac{10}{16}

sin⁡2θ=616\sin^2 \theta = \frac{6}{16}

sin⁡2θ=38\sin^2 \theta = \frac{3}{8}

Finding the Value of sin⁡θ\sin \theta

Now that we have found the value of sin⁡2θ\sin^2 \theta, we can find the value of sin⁡θ\sin \theta by taking the square root of both sides of the equation. Since sin⁡θ\sin \theta is a positive value, we take the positive square root:

sin⁡θ=38\sin \theta = \sqrt{\frac{3}{8}}

Simplifying this equation, we get:

sin⁡θ=38\sin \theta = \frac{\sqrt{3}}{\sqrt{8}}

sin⁡θ=322\sin \theta = \frac{\sqrt{3}}{2\sqrt{2}}

Rationalizing the denominator, we get:

sin⁡θ=3×22×2×2\sin \theta = \frac{\sqrt{3} \times \sqrt{2}}{2 \times \sqrt{2} \times \sqrt{2}}

sin⁡θ=64\sin \theta = \frac{\sqrt{6}}{4}

Conclusion

In this article, we used the given values of tan⁡θ=153\tan \theta=\frac{\sqrt{15}}{3} and cos⁡θ=104\cos \theta=\frac{\sqrt{10}}{4} to find the value of sin⁡θ\sin \theta. We used the Pythagorean identity to find the value of sin⁡2θ\sin^2 \theta, and then took the square root of both sides of the equation to find the value of sin⁡θ\sin \theta. The final answer is sin⁡θ=64\sin \theta = \frac{\sqrt{6}}{4}.

Additional Tips and Tricks

  • When working with trigonometric identities, it's essential to recall the definitions of the trigonometric functions and the Pythagorean identity.
  • When simplifying equations, it's crucial to rationalize the denominator to avoid any errors.
  • When finding the value of a trigonometric function, it's essential to check the quadrant in which the angle lies to ensure that the value is positive or negative.

Common Mistakes to Avoid

  • When working with trigonometric identities, it's easy to make mistakes by forgetting to square the values or by not rationalizing the denominator.
  • When finding the value of a trigonometric function, it's essential to check the quadrant in which the angle lies to ensure that the value is positive or negative.
  • When simplifying equations, it's crucial to be careful when canceling out terms to avoid any errors.

Real-World Applications

Trigonometric functions have numerous real-world applications in fields such as physics, engineering, and navigation. For example, in physics, trigonometric functions are used to describe the motion of objects in terms of their position, velocity, and acceleration. In engineering, trigonometric functions are used to design and analyze the structural integrity of buildings and bridges. In navigation, trigonometric functions are used to determine the position and course of a ship or aircraft.

Conclusion

Frequently Asked Questions

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1. This identity is used to find the value of sin⁡θ\sin \theta or cos⁡θ\cos \theta when the other value is known.

Q: How do I find the value of sin⁡θ\sin \theta using the Pythagorean identity?

A: To find the value of sin⁡θ\sin \theta using the Pythagorean identity, you need to square the value of cos⁡θ\cos \theta and subtract it from 1. Then, take the square root of the result to find the value of sin⁡θ\sin \theta.

Q: What is the difference between sin⁡θ\sin \theta and cos⁡θ\cos \theta?

A: sin⁡θ\sin \theta and cos⁡θ\cos \theta are both trigonometric functions that describe the ratios of the sides of a right-angled triangle. However, sin⁡θ\sin \theta is the ratio of the length of the side opposite the angle to the length of the hypotenuse, while cos⁡θ\cos \theta is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Q: How do I know which quadrant the angle lies in?

A: To determine which quadrant the angle lies in, you need to consider the signs of the trigonometric functions. If the angle is in the first quadrant, all three trigonometric functions are positive. If the angle is in the second quadrant, sin⁡θ\sin \theta is positive, and cos⁡θ\cos \theta and tan⁡θ\tan \theta are negative. If the angle is in the third quadrant, tan⁡θ\tan \theta is positive, and sin⁡θ\sin \theta and cos⁡θ\cos \theta are negative. If the angle is in the fourth quadrant, cos⁡θ\cos \theta is positive, and sin⁡θ\sin \theta and tan⁡θ\tan \theta are negative.

Q: What are some common mistakes to avoid when finding the value of sin⁡θ\sin \theta?

A: Some common mistakes to avoid when finding the value of sin⁡θ\sin \theta include forgetting to square the values, not rationalizing the denominator, and not checking the quadrant in which the angle lies.

Q: How do I apply the value of sin⁡θ\sin \theta to real-world problems?

A: The value of sin⁡θ\sin \theta can be applied to real-world problems in fields such as physics, engineering, and navigation. For example, in physics, sin⁡θ\sin \theta is used to describe the motion of objects in terms of their position, velocity, and acceleration. In engineering, sin⁡θ\sin \theta is used to design and analyze the structural integrity of buildings and bridges. In navigation, sin⁡θ\sin \theta is used to determine the position and course of a ship or aircraft.

Q: What are some additional tips and tricks for finding the value of sin⁡θ\sin \theta?

A: Some additional tips and tricks for finding the value of sin⁡θ\sin \theta include using the Pythagorean identity, rationalizing the denominator, and checking the quadrant in which the angle lies. Additionally, it's essential to be careful when canceling out terms and to double-check the calculations to avoid any errors.

Conclusion

In conclusion, finding the value of sin⁡θ\sin \theta using the Pythagorean identity requires a thorough understanding of trigonometric identities and the Pythagorean identity. By following the steps outlined in this article and avoiding common mistakes, we can find the value of sin⁡θ\sin \theta and apply it to real-world problems.