Given That $\cos \theta=\frac{8}{17}$ And That $\theta$ Lies In Quadrant IV, What Is The Exact Value Of $\sin 2 \theta$?A. $\frac{30}{17}$ B. $\frac{240}{289}$ C. $-\frac{30}{17}$ D.

Feb 26, 2025 by ADMIN 186 views

Solving Trigonometric Equations: Finding the Exact Value of $\sin 2 \theta$

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on solving trigonometric equations, specifically finding the exact value of $\sin 2 \theta$ given that $\cos \theta = \frac{8}{17}$ and that $\theta$ lies in Quadrant IV.

To solve this problem, we need to recall the double-angle identity for sine, which is given by:

\sin 2 \theta = 2 \sin \theta \cos \theta

We are given that $\cos \theta = \frac{8}{17}$ and that $\theta$ lies in Quadrant IV. This means that $\sin \theta$ is negative, since the sine function is negative in Quadrant IV.

Finding the Value of $\sin \theta$

To find the value of $\sin \theta$ , we can use the Pythagorean identity, which is given by:

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

Since we are given that $\cos \theta = \frac{8}{17}$ , we can substitute this value into the Pythagorean identity to get:

\sin^2 \theta + \left(\frac{8}{17}\right)^2 = 1

Simplifying this equation, we get:

\sin^2 \theta + \frac{64}{289} = 1

Subtracting $\frac{64}{289}$ from both sides, we get:

\sin^2 \theta = \frac{225}{289}

Taking the square root of both sides, we get:

\sin \theta = -\frac{15}{17}

Finding the Value of $\sin 2 \theta$

Now that we have found the value of $\sin \theta$ , we can use the double-angle identity for sine to find the value of $\sin 2 \theta$ . Substituting the values of $\sin \theta$ and $\cos \theta$ into the double-angle identity, we get:

\sin 2 \theta = 2 \left(-\frac{15}{17}\right) \left(\frac{8}{17}\right)

Simplifying this expression, we get:

\sin 2 \theta = -\frac{240}{289}

In this article, we have solved a trigonometric equation to find the exact value of $\sin 2 \theta$ given that $\cos \theta = \frac{8}{17}$ and that $\theta$ lies in Quadrant IV. We used the Pythagorean identity to find the value of $\sin \theta$ , and then used the double-angle identity for sine to find the value of $\sin 2 \theta$ . The final answer is:

\sin 2 \theta = -\frac{240}{289}

This result is consistent with the options provided in the problem.

This problem is a classic example of a trigonometric equation that requires the use of the Pythagorean identity and the double-angle identity for sine. The solution involves a series of algebraic manipulations and trigonometric identities, which are essential skills for any student of mathematics.

In conclusion, this problem demonstrates the importance of understanding trigonometric identities and how to apply them to solve complex equations. With practice and patience, students can develop the skills and confidence needed to tackle even the most challenging trigonometric problems.

For students who want to learn more about trigonometry, there are many online resources available, including:

Khan Academy: Trigonometry
MIT OpenCourseWare: Trigonometry
Wolfram Alpha: Trigonometry

These resources provide a wealth of information and practice problems to help students master the subject.

In conclusion, solving trigonometric equations is an essential skill for any student of mathematics. By understanding the Pythagorean identity and the double-angle identity for sine, students can tackle even the most complex equations with confidence. With practice and patience, students can develop the skills and knowledge needed to succeed in mathematics and beyond.
Trigonometry Q&A: Common Questions and Answers

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will answer some common questions and provide explanations for various trigonometric concepts.

Q: What is the difference between sine, cosine, and tangent?

A: Sine, cosine, and tangent are three fundamental trigonometric functions that are used to describe the relationships between the sides and angles of triangles. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, and the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

Q: How do I remember the order of the trigonometric functions?

A: One way to remember the order of the trigonometric functions is to use the mnemonic "SOH-CAH-TOA". This stands for:

Sine = Opposite over Hypotenuse
Cosine = Adjacent over Hypotenuse
Tangent = Opposite over Adjacent

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental trigonometric identity that states:

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

This identity is used to find the value of one trigonometric function if the value of another is known.

Q: What is the double-angle identity for sine?

A: The double-angle identity for sine is a fundamental trigonometric identity that states:

\sin 2 \theta = 2 \sin \theta \cos \theta

This identity is used to find the value of the sine of a double angle if the values of the sine and cosine of the original angle are known.

Q: How do I solve trigonometric equations?

A: To solve trigonometric equations, you need to use the trigonometric identities and formulas to isolate the variable. Here are the steps to follow:

Simplify the equation by using the trigonometric identities and formulas.
Isolate the variable by using algebraic manipulations.
Check the solution by plugging it back into the original equation.

Q: What is the difference between a right triangle and an oblique triangle?

A: A right triangle is a triangle with one right angle (90 degrees), while an oblique triangle is a triangle with no right angles. Right triangles are used to define the trigonometric functions, while oblique triangles are used to solve problems involving triangles with no right angles.

Q: How do I use trigonometry in real-life situations?

A: Trigonometry is used in a variety of real-life situations, including:

Navigation: Trigonometry is used to calculate distances and directions between two points on the Earth's surface.
Physics: Trigonometry is used to describe the motion of objects in terms of their position, velocity, and acceleration.
Engineering: Trigonometry is used to design and build structures such as bridges, buildings, and roads.
Computer graphics: Trigonometry is used to create 3D models and animations.

In this article, we have answered some common questions and provided explanations for various trigonometric concepts. We hope that this article has been helpful in clarifying any doubts you may have had about trigonometry. If you have any further questions or need additional help, please don't hesitate to ask.