If $\cos \theta=\frac{3}{4}$ And $\csc \theta\ \textgreater \ 0$, Find $ Tan Θ \tan \theta Tan Θ [/tex].
Introduction
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 value of tan theta given the values of cos theta and csc theta.
Understanding the Given Information
We are given that cos theta = 3/4 and csc theta > 0. The value of csc theta is the reciprocal of sin theta, which means csc theta = 1/sin theta. Since csc theta > 0, we know that sin theta > 0.
Recalling Trigonometric Identities
To solve this problem, we need to recall some basic trigonometric identities. The Pythagorean identity states that sin^2 theta + cos^2 theta = 1. We can use this identity to find the value of sin theta.
Finding Sin Theta
Using the Pythagorean identity, we can find the value of sin theta as follows:
sin^2 theta + cos^2 theta = 1 sin^2 theta + (3/4)^2 = 1 sin^2 theta + 9/16 = 1 sin^2 theta = 1 - 9/16 sin^2 theta = 7/16 sin theta = sqrt(7/16) sin theta = sqrt(7)/4
Finding Tan Theta
Now that we have found the value of sin theta, we can find the value of tan theta using the identity tan theta = sin theta / cos theta.
tan theta = sin theta / cos theta tan theta = (sqrt(7)/4) / (3/4) tan theta = sqrt(7)/3
Conclusion
In this article, we have solved a trigonometric equation to find the value of tan theta given the values of cos theta and csc theta. We have used the Pythagorean identity to find the value of sin theta and then used the identity tan theta = sin theta / cos theta to find the value of tan theta. The final answer is tan theta = sqrt(7)/3.
Additional Tips and Tricks
- When solving trigonometric equations, it is essential to recall the basic trigonometric identities, such as the Pythagorean identity.
- Use the given information to find the value of one trigonometric function, and then use that value to find the value of another trigonometric function.
- Be careful when simplifying expressions and make sure to check your work.
Common Mistakes to Avoid
- Make sure to recall the basic trigonometric identities and use them correctly.
- Be careful when simplifying expressions and make sure to check your work.
- Avoid making assumptions about the value of a trigonometric function without using the given information.
Real-World Applications
Trigonometry has numerous real-world applications, including:
- Navigation: Trigonometry is used in navigation to calculate distances and directions.
- Physics: Trigonometry is used in physics to describe the motion of objects.
- Engineering: Trigonometry is used in engineering to design and build structures.
- Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.
Final Thoughts
Introduction
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 frequently asked questions about trigonometry.
Q: What is the difference between sine, cosine, and tangent?
A: Sine, cosine, and tangent are three fundamental trigonometric functions. Sine is the ratio of the length of the opposite side to the length of the hypotenuse, cosine is the ratio of the length of the adjacent side to the length of the hypotenuse, and tangent 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 can be used to find the value of one trigonometric function if the values of the other two functions are known.
Q: How do I find the value of a trigonometric function if I know the values of the other two functions?
A: To find the value of a trigonometric function if you know the values of the other two functions, you can use the Pythagorean identity. For example, if you know the values of sin theta and cos theta, you can use the Pythagorean identity to find the value of tan theta.
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). An oblique triangle is a triangle with no right angles.
Q: How do I find the value of a trigonometric function in an oblique triangle?
A: To find the value of a trigonometric function in an oblique triangle, you can use the law of sines or the law of cosines. The law of sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles of a triangle. The law of cosines states that the square of the length of a side is equal to the sum of the squares of the lengths of the other two sides minus twice the product of those two sides times the cosine of the angle between them.
Q: What is the law of sines?
A: The law of sines is a fundamental trigonometric identity that states:
a / sin(A) = b / sin(B) = c / sin(C)
where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the measures of the angles opposite those sides.
Q: What is the law of cosines?
A: The law of cosines is a fundamental trigonometric identity that states:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the length of the side opposite angle C, and a and b are the lengths of the other two sides.
Q: How do I use the law of sines and the law of cosines to find the value of a trigonometric function?
A: To use the law of sines and the law of cosines to find the value of a trigonometric function, you can rearrange the formulas to solve for the desired function. For example, if you know the values of the lengths of the sides of a triangle and the measure of one angle, you can use the law of sines to find the value of the sine of that angle.
Conclusion
Trigonometry is a fundamental subject that has numerous applications in various fields. By understanding the basic trigonometric functions, identities, and formulas, you can solve a wide range of problems involving triangles and angles. We hope this article has been helpful in answering some of your frequently asked questions about trigonometry.