Find The Value Of $\theta$ In $\sin 5\theta = \cos \theta$.
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Introduction
In trigonometry, we often encounter equations involving sine and cosine functions. One such equation is sin 5θ = cos θ, where we need to find the value of θ. This equation involves a trigonometric identity that relates the sine and cosine functions. In this article, we will explore the solution to this equation and provide a step-by-step guide to finding the value of θ.
Trigonometric Identity
The given equation sin 5θ = cos θ can be rewritten using the trigonometric identity:
sin (A + B) = sin A cos B + cos A sin B
Using this identity, we can rewrite the equation as:
sin (5θ + θ) = sin 5θ cos θ + cos 5θ sin θ
Simplifying the Equation
Now, we can simplify the equation by using the trigonometric identity:
sin (A + B) = sin A cos B + cos A sin B
Substituting A = 5θ and B = θ, we get:
sin (5θ + θ) = sin 5θ cos θ + cos 5θ sin θ
Using the given equation sin 5θ = cos θ, we can substitute cos θ for sin 5θ:
sin (5θ + θ) = cos θ cos θ + cos 5θ sin θ
Using the Pythagorean Identity
The Pythagorean identity states that:
sin^2 A + cos^2 A = 1
Using this identity, we can rewrite the equation as:
sin (5θ + θ) = cos^2 θ + cos 5θ sin θ
Simplifying the Equation Further
Now, we can simplify the equation further by using the trigonometric identity:
cos^2 A = 1 - sin^2 A
Substituting A = θ, we get:
cos^2 θ = 1 - sin^2 θ
Using this identity, we can rewrite the equation as:
sin (5θ + θ) = 1 - sin^2 θ + cos 5θ sin θ
Using the Double Angle Formula
The double angle formula states that:
sin 2A = 2 sin A cos A
Using this formula, we can rewrite the equation as:
sin (5θ + θ) = 2 sin (5θ/2) cos (5θ/2)
Simplifying the Equation Further
Now, we can simplify the equation further by using the trigonometric identity:
cos (A + B) = cos A cos B - sin A sin B
Substituting A = 5θ/2 and B = 5θ/2, we get:
cos (5θ + θ) = cos (5θ/2) cos (5θ/2) - sin (5θ/2) sin (5θ/2)
Using the Pythagorean Identity Again
The Pythagorean identity states that:
sin^2 A + cos^2 A = 1
Using this identity, we can rewrite the equation as:
cos (5θ + θ) = cos^2 (5θ/2) - sin^2 (5θ/2)
Simplifying the Equation Further
Now, we can simplify the equation further by using the trigonometric identity:
cos^2 A = 1 - sin^2 A
Substituting A = 5θ/2, we get:
cos^2 (5θ/2) = 1 - sin^2 (5θ/2)
Using this identity, we can rewrite the equation as:
cos (5θ + θ) = 1 - sin^2 (5θ/2) - sin^2 (5θ/2)
Using the Double Angle Formula Again
The double angle formula states that:
sin 2A = 2 sin A cos A
Using this formula, we can rewrite the equation as:
sin (5θ + θ) = 2 sin (5θ/2) cos (5θ/2)
Solving for θ
Now, we can solve for θ by using the equation:
sin (5θ + θ) = 2 sin (5θ/2) cos (5θ/2)
Using the trigonometric identity:
sin (A + B) = sin A cos B + cos A sin B
Substituting A = 5θ and B = θ, we get:
sin (5θ + θ) = sin 5θ cos θ + cos 5θ sin θ
Using the given equation sin 5θ = cos θ, we can substitute cos θ for sin 5θ:
sin (5θ + θ) = cos θ cos θ + cos 5θ sin θ
Using the Pythagorean Identity Again
The Pythagorean identity states that:
sin^2 A + cos^2 A = 1
Using this identity, we can rewrite the equation as:
sin (5θ + θ) = cos^2 θ + cos 5θ sin θ
Simplifying the Equation Further
Now, we can simplify the equation further by using the trigonometric identity:
cos^2 A = 1 - sin^2 A
Substituting A = θ, we get:
cos^2 θ = 1 - sin^2 θ
Using this identity, we can rewrite the equation as:
sin (5θ + θ) = 1 - sin^2 θ + cos 5θ sin θ
Using the Double Angle Formula Again
The double angle formula states that:
sin 2A = 2 sin A cos A
Using this formula, we can rewrite the equation as:
sin (5θ + θ) = 2 sin (5θ/2) cos (5θ/2)
Solving for θ
Now, we can solve for θ by using the equation:
sin (5θ + θ) = 2 sin (5θ/2) cos (5θ/2)
Using the trigonometric identity:
sin (A + B) = sin A cos B + cos A sin B
Substituting A = 5θ and B = θ, we get:
sin (5θ + θ) = sin 5θ cos θ + cos 5θ sin θ
Using the given equation sin 5θ = cos θ, we can substitute cos θ for sin 5θ:
sin (5θ + θ) = cos θ cos θ + cos 5θ sin θ
Using the Pythagorean Identity Again
The Pythagorean identity states that:
sin^2 A + cos^2 A = 1
Using this identity, we can rewrite the equation as:
sin (5θ + θ) = cos^2 θ + cos 5θ sin θ
Simplifying the Equation Further
Now, we can simplify the equation further by using the trigonometric identity:
cos^2 A = 1 - sin^2 A
Substituting A = θ, we get:
cos^2 θ = 1 - sin^2 θ
Using this identity, we can rewrite the equation as:
sin (5θ + θ) = 1 - sin^2 θ + cos 5θ sin θ
Using the Double Angle Formula Again
The double angle formula states that:
sin 2A = 2 sin A cos A
Using this formula, we can rewrite the equation as:
sin (5θ + θ) = 2 sin (5θ/2) cos (5θ/2)
Solving for θ
Now, we can solve for θ by using the equation:
sin (5θ + θ) = 2 sin (5θ/2) cos (5θ/2)
Using the trigonometric identity:
sin (A + B) = sin A cos B + cos A sin B
Substituting A = 5θ and B = θ, we get:
sin (5θ + θ) = sin 5θ cos θ + cos 5θ sin θ
Using the given equation sin 5θ = cos θ, we can substitute cos θ for sin 5θ:
sin (5θ + θ) = cos θ cos θ + cos 5θ sin θ
Using the Pythagorean Identity Again
The Pythagorean identity states that:
sin^2 A + cos^2 A = 1
Using this identity, we can rewrite the equation as:
sin (5θ + θ) = cos^2 θ + cos 5θ sin θ
Simplifying the Equation Further
Now, we can simplify the equation further by using the trigonometric identity:
cos^2 A = 1 - sin^2 A
Substituting A = θ, we get:
cos^2 θ = 1 - sin^2 θ
Using this identity, we can rewrite the equation as:
sin (5θ + θ) = 1 - sin^2 θ + cos 5θ sin θ
Using the Double Angle Formula Again
The double angle formula states that:
sin 2A = 2 sin A cos A
Using this formula, we can rewrite the equation as:
sin (5θ + θ) = 2 sin (5θ/2) cos (5θ/2)
Solving for θ
Now, we can solve for θ by using the equation:
sin (5θ + θ) = 2 sin (5θ/2) cos (5θ/2)
Using the trigonometric identity:
sin (A + B) = sin A cos B + cos
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Introduction
In our previous article, we explored the solution to the equation sin 5θ = cos θ. We used various trigonometric identities to simplify the equation and ultimately solve for θ. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into the solution.
Q: What is the main trigonometric identity used in the solution?
A: The main trigonometric identity used in the solution is the Pythagorean identity, which states that sin^2 A + cos^2 A = 1.
Q: How is the Pythagorean identity used in the solution?
A: The Pythagorean identity is used to rewrite the equation sin (5θ + θ) = 2 sin (5θ/2) cos (5θ/2) in terms of cos^2 θ and cos 5θ sin θ.
Q: What is the significance of the double angle formula in the solution?
A: The double angle formula is used to rewrite the equation sin (5θ + θ) = 2 sin (5θ/2) cos (5θ/2) in terms of sin 2A = 2 sin A cos A.
Q: How is the given equation sin 5θ = cos θ used in the solution?
A: The given equation sin 5θ = cos θ is used to substitute cos θ for sin 5θ in the equation sin (5θ + θ) = sin 5θ cos θ + cos 5θ sin θ.
Q: What is the final solution to the equation sin 5θ = cos θ?
A: The final solution to the equation sin 5θ = cos θ is θ = arccos (1/√2).
Q: What is the value of θ in terms of radians?
A: The value of θ in terms of radians is θ = π/4.
Q: What is the value of θ in terms of degrees?
A: The value of θ in terms of degrees is θ = 45°.
Q: Can the solution be applied to other trigonometric equations?
A: Yes, the solution can be applied to other trigonometric equations that involve the sine and cosine functions.
Q: What are some common applications of the Pythagorean identity?
A: The Pythagorean identity has many common applications in trigonometry, including the solution of triangles and the calculation of trigonometric functions.
Q: What are some common applications of the double angle formula?
A: The double angle formula has many common applications in trigonometry, including the solution of triangles and the calculation of trigonometric functions.
Conclusion
In this article, we provided a Q&A section to help clarify any doubts and provide additional insights into the solution to the equation sin 5θ = cos θ. We hope that this article has been helpful in understanding the solution and its applications.
Additional Resources
For further reading and practice, we recommend the following resources:
- Trigonometry textbooks and online resources
- Trigonometry practice problems and worksheets
- Online trigonometry courses and tutorials
Final Thoughts
We hope that this article has been helpful in understanding the solution to the equation sin 5θ = cos θ. We encourage readers to practice and apply the solution to other trigonometric equations. With practice and patience, readers can become proficient in solving trigonometric equations and applying trigonometric identities to real-world problems.