What Is Sin ⁡ Θ \sin \theta Sin Θ If Cos ⁡ Θ = 0.8 \cos \theta = 0.8 Cos Θ = 0.8 And Θ \theta Θ Is In Quadrant I?

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What is sin⁡θ\sin \theta if cos⁡θ=0.8\cos \theta = 0.8 and θ\theta is in Quadrant I?

In trigonometry, the sine and cosine functions are two fundamental concepts that are used to describe the relationships between the angles and side lengths of triangles. The sine function, denoted by sin⁡θ\sin \theta, is defined as the ratio of the length of the side opposite the angle θ\theta to the length of the hypotenuse in a right-angled triangle. On the other hand, the cosine function, denoted by cos⁡θ\cos \theta, is defined as the ratio of the length of the side adjacent to the angle θ\theta to the length of the hypotenuse. In this article, we will explore the concept of sin⁡θ\sin \theta and how to find its value when cos⁡θ=0.8\cos \theta = 0.8 and θ\theta is in Quadrant I.

Understanding Quadrant I

Quadrant I is one of the four quadrants in the Cartesian coordinate system, where both the x-coordinate and the y-coordinate are positive. In the context of trigonometry, Quadrant I refers to the region of the unit circle where the angle θ\theta is measured counterclockwise from the positive x-axis. When θ\theta is in Quadrant I, the sine function is positive, and the cosine function is also positive.

The Pythagorean Identity

The Pythagorean identity is a fundamental relationship between the sine and cosine functions, which states that:

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

This identity can be used to find the value of sin⁡θ\sin \theta when cos⁡θ\cos \theta is known. By rearranging the equation, we can solve for sin⁡θ\sin \theta:

sin⁡θ=±1−cos⁡2θ\sin \theta = \pm \sqrt{1 - \cos^2 \theta}

Since θ\theta is in Quadrant I, we know that sin⁡θ\sin \theta is positive. Therefore, we can take the positive square root:

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

Finding sin⁡θ\sin \theta

Now that we have the formula for sin⁡θ\sin \theta, we can substitute the given value of cos⁡θ=0.8\cos \theta = 0.8 into the equation:

sin⁡θ=1−(0.8)2\sin \theta = \sqrt{1 - (0.8)^2}

sin⁡θ=1−0.64\sin \theta = \sqrt{1 - 0.64}

sin⁡θ=0.36\sin \theta = \sqrt{0.36}

sin⁡θ=0.6\sin \theta = 0.6

Therefore, the value of sin⁡θ\sin \theta is 0.6 when cos⁡θ=0.8\cos \theta = 0.8 and θ\theta is in Quadrant I.

In conclusion, we have explored the concept of sin⁡θ\sin \theta and how to find its value when cos⁡θ=0.8\cos \theta = 0.8 and θ\theta is in Quadrant I. We used the Pythagorean identity to derive a formula for sin⁡θ\sin \theta and then substituted the given value of cos⁡θ\cos \theta into the equation to find the value of sin⁡θ\sin \theta. The result is that sin⁡θ=0.6\sin \theta = 0.6 when cos⁡θ=0.8\cos \theta = 0.8 and θ\theta is in Quadrant I.

Here are a few additional examples of how to find sin⁡θ\sin \theta when cos⁡θ\cos \theta is known:

  • If cos⁡θ=0.5\cos \theta = 0.5 and θ\theta is in Quadrant I, then sin⁡θ=1−(0.5)2=0.75=0.866\sin \theta = \sqrt{1 - (0.5)^2} = \sqrt{0.75} = 0.866.
  • If cos⁡θ=0.2\cos \theta = 0.2 and θ\theta is in Quadrant I, then sin⁡θ=1−(0.2)2=0.96=0.979\sin \theta = \sqrt{1 - (0.2)^2} = \sqrt{0.96} = 0.979.

These examples demonstrate how to use the Pythagorean identity to find the value of sin⁡θ\sin \theta when cos⁡θ\cos \theta is known.

The concept of sin⁡θ\sin \theta has numerous real-world applications in fields such as physics, engineering, and computer science. For example:

  • In physics, the sine function is used to describe the motion of objects in circular motion, such as the rotation of a wheel or the orbit of a planet.
  • In engineering, the sine function is used to design and analyze systems that involve circular motion, such as gears and pulleys.
  • In computer science, the sine function is used in algorithms for graphics rendering and game development.

These applications demonstrate the importance of the sine function in real-world problems and highlight the need for a thorough understanding of trigonometry.

In conclusion, we have explored the concept of sin⁡θ\sin \theta and how to find its value when cos⁡θ=0.8\cos \theta = 0.8 and θ\theta is in Quadrant I. We used the Pythagorean identity to derive a formula for sin⁡θ\sin \theta and then substituted the given value of cos⁡θ\cos \theta into the equation to find the value of sin⁡θ\sin \theta. The result is that sin⁡θ=0.6\sin \theta = 0.6 when cos⁡θ=0.8\cos \theta = 0.8 and θ\theta is in Quadrant I. We also discussed the real-world applications of the sine function and provided additional examples of how to find sin⁡θ\sin \theta when cos⁡θ\cos \theta is known.
Q&A: What is sin⁡θ\sin \theta if cos⁡θ=0.8\cos \theta = 0.8 and θ\theta is in Quadrant I?

Here are some frequently asked questions about the concept of sin⁡θ\sin \theta and how to find its value when cos⁡θ=0.8\cos \theta = 0.8 and θ\theta is in Quadrant I.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental relationship between the sine and cosine functions, which states that:

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

This identity can be used to find the value of sin⁡θ\sin \theta when cos⁡θ\cos \theta is known.

Q: How do I find sin⁡θ\sin \theta when cos⁡θ\cos \theta is known?

A: To find sin⁡θ\sin \theta when cos⁡θ\cos \theta is known, you can use the Pythagorean identity:

sin⁡θ=±1−cos⁡2θ\sin \theta = \pm \sqrt{1 - \cos^2 \theta}

Since θ\theta is in Quadrant I, we know that sin⁡θ\sin \theta is positive. Therefore, we can take the positive square root:

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

Q: What if cos⁡θ\cos \theta is negative?

A: If cos⁡θ\cos \theta is negative, then sin⁡θ\sin \theta will also be negative. In this case, you can use the Pythagorean identity:

sin⁡θ=±1−cos⁡2θ\sin \theta = \pm \sqrt{1 - \cos^2 \theta}

Since sin⁡θ\sin \theta is negative, you can take the negative square root:

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

Q: Can I use the Pythagorean identity to find cos⁡θ\cos \theta when sin⁡θ\sin \theta is known?

A: Yes, you can use the Pythagorean identity to find cos⁡θ\cos \theta when sin⁡θ\sin \theta is known. Simply rearrange the equation:

cos⁡θ=±1−sin⁡2θ\cos \theta = \pm \sqrt{1 - \sin^2 \theta}

Since θ\theta is in Quadrant I, we know that cos⁡θ\cos \theta is positive. Therefore, we can take the positive square root:

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

Q: What if I have a value for tan⁡θ\tan \theta instead of cos⁡θ\cos \theta?

A: If you have a value for tan⁡θ\tan \theta instead of cos⁡θ\cos \theta, you can use the following relationship:

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

Rearrange the equation to solve for sin⁡θ\sin \theta:

sin⁡θ=tan⁡θcos⁡θ\sin \theta = \tan \theta \cos \theta

Now you can use the Pythagorean identity to find sin⁡θ\sin \theta:

sin⁡θ=tan⁡θcos⁡θ=tan⁡θ1−sin⁡2θ\sin \theta = \tan \theta \cos \theta = \tan \theta \sqrt{1 - \sin^2 \theta}

Q: Can I use a calculator to find sin⁡θ\sin \theta?

A: Yes, you can use a calculator to find sin⁡θ\sin \theta. Simply enter the value of cos⁡θ\cos \theta and the calculator will give you the value of sin⁡θ\sin \theta.

In conclusion, we have answered some frequently asked questions about the concept of sin⁡θ\sin \theta and how to find its value when cos⁡θ=0.8\cos \theta = 0.8 and θ\theta is in Quadrant I. We have discussed the Pythagorean identity, how to find sin⁡θ\sin \theta when cos⁡θ\cos \theta is known, and how to use a calculator to find sin⁡θ\sin \theta. We hope this article has been helpful in understanding the concept of sin⁡θ\sin \theta and how to find its value in different scenarios.