What Is The Value Of Sin ⁡ Θ \sin \theta Sin Θ , Cos ⁡ Θ \cos \theta Cos Θ , And Tan ⁡ Θ \tan \theta Tan Θ , Given That ( − 6 , 3 (-6,3 ( − 6 , 3 ] Is A Point On The Terminal Side Of Θ \theta Θ ?Drag And Drop Your Answers To Correctly Match The Trigonometric Functions

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Introduction

In trigonometry, the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta are essential in solving various problems. These values can be determined using the coordinates of a point on the terminal side of an angle. In this article, we will explore how to find the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta given that (−6,3)(-6,3) is a point on the terminal side of θ\theta.

Understanding the Trigonometric Functions

Before we proceed, let's briefly review the trigonometric functions.

  • Sine: The sine of an angle θ\theta is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
  • Cosine: The cosine of an angle θ\theta is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
  • Tangent: The tangent of an angle θ\theta is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle.

Finding the Values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta

To find the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta, we need to use the coordinates of the point (−6,3)(-6,3) on the terminal side of θ\theta. We can use the following formulas:

  • sin⁡θ=yr\sin \theta = \frac{y}{r}
  • cos⁡θ=xr\cos \theta = \frac{x}{r}
  • tan⁡θ=yx\tan \theta = \frac{y}{x}

where xx and yy are the coordinates of the point, and rr is the distance from the origin to the point.

Calculating the Distance from the Origin to the Point

To calculate the distance from the origin to the point (−6,3)(-6,3), we can use the distance formula:

r=x2+y2r = \sqrt{x^2 + y^2}

Substituting the values of xx and yy, we get:

r=(−6)2+32r = \sqrt{(-6)^2 + 3^2} r=36+9r = \sqrt{36 + 9} r=45r = \sqrt{45}

Calculating the Values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta

Now that we have the value of rr, we can calculate the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta using the formulas:

  • sin⁡θ=yr=345\sin \theta = \frac{y}{r} = \frac{3}{\sqrt{45}}
  • cos⁡θ=xr=−645\cos \theta = \frac{x}{r} = \frac{-6}{\sqrt{45}}
  • tan⁡θ=yx=3−6=−12\tan \theta = \frac{y}{x} = \frac{3}{-6} = -\frac{1}{2}

Simplifying the Values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta

To simplify the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta, we can rationalize the denominators by multiplying the numerator and denominator by 45\sqrt{45}:

  • sin⁡θ=345×4545=34545\sin \theta = \frac{3}{\sqrt{45}} \times \frac{\sqrt{45}}{\sqrt{45}} = \frac{3\sqrt{45}}{45}
  • cos⁡θ=−645×4545=−64545\cos \theta = \frac{-6}{\sqrt{45}} \times \frac{\sqrt{45}}{\sqrt{45}} = \frac{-6\sqrt{45}}{45}
  • tan⁡θ=−12\tan \theta = -\frac{1}{2} (no simplification needed)

Conclusion

In this article, we have explored how to find the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta given that (−6,3)(-6,3) is a point on the terminal side of θ\theta. We have used the coordinates of the point and the distance formula to calculate the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta. We have also simplified the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta by rationalizing the denominators.

Drag and Drop Your Answers

Here are the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta:

  • sin⁡θ=34545\sin \theta = \frac{3\sqrt{45}}{45}
  • cos⁡θ=−64545\cos \theta = \frac{-6\sqrt{45}}{45}
  • tan⁡θ=−12\tan \theta = -\frac{1}{2}

Drag and drop your answers to correctly match the trigonometric functions.

Correct Answers

  • sin⁡θ=34545\sin \theta = \frac{3\sqrt{45}}{45}
  • cos⁡θ=−64545\cos \theta = \frac{-6\sqrt{45}}{45}
  • tan⁡θ=−12\tan \theta = -\frac{1}{2}

Q&A: Finding the Values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta

Q: What are the trigonometric functions?

A: The trigonometric functions are:

  • Sine: The sine of an angle θ\theta is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
  • Cosine: The cosine of an angle θ\theta is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
  • Tangent: The tangent of an angle θ\theta is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle.

Q: How do I find the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta?

A: To find the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta, you need to use the coordinates of the point on the terminal side of θ\theta. You can use the following formulas:

  • sin⁡θ=yr\sin \theta = \frac{y}{r}
  • cos⁡θ=xr\cos \theta = \frac{x}{r}
  • tan⁡θ=yx\tan \theta = \frac{y}{x}

where xx and yy are the coordinates of the point, and rr is the distance from the origin to the point.

Q: How do I calculate the distance from the origin to the point?

A: To calculate the distance from the origin to the point, you can use the distance formula:

r=x2+y2r = \sqrt{x^2 + y^2}

Q: What is the value of rr in this problem?

A: In this problem, the value of rr is:

r=(−6)2+32r = \sqrt{(-6)^2 + 3^2} r=36+9r = \sqrt{36 + 9} r=45r = \sqrt{45}

Q: How do I calculate the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta?

A: Now that we have the value of rr, we can calculate the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta using the formulas:

  • sin⁡θ=yr=345\sin \theta = \frac{y}{r} = \frac{3}{\sqrt{45}}
  • cos⁡θ=xr=−645\cos \theta = \frac{x}{r} = \frac{-6}{\sqrt{45}}
  • tan⁡θ=yx=3−6=−12\tan \theta = \frac{y}{x} = \frac{3}{-6} = -\frac{1}{2}

Q: How do I simplify the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta?

A: To simplify the values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta, you can rationalize the denominators by multiplying the numerator and denominator by 45\sqrt{45}:

  • sin⁡θ=345×4545=34545\sin \theta = \frac{3}{\sqrt{45}} \times \frac{\sqrt{45}}{\sqrt{45}} = \frac{3\sqrt{45}}{45}
  • cos⁡θ=−645×4545=−64545\cos \theta = \frac{-6}{\sqrt{45}} \times \frac{\sqrt{45}}{\sqrt{45}} = \frac{-6\sqrt{45}}{45}
  • tan⁡θ=−12\tan \theta = -\frac{1}{2} (no simplification needed)

Q: What are the final values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta?

A: The final values of sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta are:

  • sin⁡θ=34545\sin \theta = \frac{3\sqrt{45}}{45}
  • cos⁡θ=−64545\cos \theta = \frac{-6\sqrt{45}}{45}
  • tan⁡θ=−12\tan \theta = -\frac{1}{2}

Q: How do I use these values in a problem?

A: You can use these values in a problem by substituting them into the relevant trigonometric function. For example, if you are given a right-angled triangle with an angle θ\theta and you want to find the length of the opposite side, you can use the value of sin⁡θ\sin \theta to find the length of the opposite side.

Q: What are some common applications of trigonometry?

A: Trigonometry has many common applications in various fields, including:

  • Physics: Trigonometry is used to describe the motion of objects in terms of position, velocity, and acceleration.
  • Engineering: Trigonometry is used to design and build structures such as bridges, buildings, and roads.
  • Navigation: Trigonometry is used to determine the position and direction of objects on the Earth's surface.
  • Computer Science: Trigonometry is used in computer graphics and game development to create 3D models and animations.

Q: What are some common mistakes to avoid when working with trigonometry?

A: Some common mistakes to avoid when working with trigonometry include:

  • Not using the correct formula: Make sure to use the correct formula for the trigonometric function you are working with.
  • Not simplifying the expression: Make sure to simplify the expression before solving for the unknown value.
  • Not checking the units: Make sure to check the units of the answer to ensure that they are correct.

Q: What are some resources for learning more about trigonometry?

A: Some resources for learning more about trigonometry include:

  • Textbooks: There are many textbooks available that cover trigonometry in detail.
  • Online resources: There are many online resources available that provide tutorials, examples, and practice problems for trigonometry.
  • Video lectures: There are many video lectures available that cover trigonometry in detail.
  • Practice problems: Practice problems are an essential part of learning trigonometry. Make sure to practice solving problems regularly to build your skills and confidence.