DetailsPg. 420: Problems #5, 6, 8, 9, 17, 20-22, 28, 295. What Is $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$?6. What Is $\tan^{-1}(\sqrt{3})$?7. What Are All The Angles (in Degrees) That Have A Cosine Value Of 0.74?8. What Are All

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Details of Trigonometric Functions: Problems #5, 6, 8, 9, 17, 20-22, 28, 29

Understanding the Basics of Trigonometric Functions

Trigonometric functions are a crucial part of mathematics, and they play a vital role in various mathematical and scientific applications. These functions are used to describe the relationships between the sides and angles of triangles. In this article, we will focus on solving problems related to trigonometric functions, specifically the inverse trigonometric functions.

Problem #5: Finding the Value of $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$

The inverse sine function, denoted by $\sin^{-1}x$, is the angle whose sine is equal to x. In this problem, we need to find the value of $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$.

To solve this problem, we need to recall the unit circle and the values of sine and cosine functions for common angles. We know that the sine of 45 degrees is $\frac{\sqrt{2}}{2}$.

Using this information, we can conclude that $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = 45^{\circ}$.

Problem #6: Finding the Value of $\tan^{-1}(\sqrt{3})$

The inverse tangent function, denoted by $\tan^{-1}x$, is the angle whose tangent is equal to x. In this problem, we need to find the value of $\tan^{-1}(\sqrt{3})$.

To solve this problem, we need to recall the unit circle and the values of tangent and cotangent functions for common angles. We know that the tangent of 60 degrees is $\sqrt{3}$.

Using this information, we can conclude that $\tan^{-1}(\sqrt{3}) = 60^{\circ}$.

Problem #8: Finding the Angles with a Cosine Value of 0.74

The cosine function is used to describe the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In this problem, we need to find all the angles whose cosine value is 0.74.

To solve this problem, we need to use the inverse cosine function, denoted by $\cos^{-1}x$, which is the angle whose cosine is equal to x. We can use a calculator or a trigonometric table to find the angle whose cosine is 0.74.

Using a calculator, we find that $\cos^{-1}(0.74) = 41.41^{\circ}$ and $\cos^{-1}(0.74) = 138.59^{\circ}$.

Problem #9: Finding the Angles with a Sine Value of 0.5

The sine function is used to describe the ratio of the opposite side to the hypotenuse in a right-angled triangle. In this problem, we need to find all the angles whose sine value is 0.5.

To solve this problem, we need to use the inverse sine function, denoted by $\sin^{-1}x$, which is the angle whose sine is equal to x. We can use a calculator or a trigonometric table to find the angle whose sine is 0.5.

Using a calculator, we find that $\sin^{-1}(0.5) = 30^{\circ}$ and $\sin^{-1}(0.5) = 150^{\circ}$.

Problem #17: Finding the Angles with a Tangent Value of 1

The tangent function is used to describe the ratio of the opposite side to the adjacent side in a right-angled triangle. In this problem, we need to find all the angles whose tangent value is 1.

To solve this problem, we need to use the inverse tangent function, denoted by $\tan^{-1}x$, which is the angle whose tangent is equal to x. We can use a calculator or a trigonometric table to find the angle whose tangent is 1.

Using a calculator, we find that $\tan^{-1}(1) = 45^{\circ}$ and $\tan^{-1}(1) = 225^{\circ}$.

Problem #20-22: Finding the Angles with a Cosine Value of 0.5

The cosine function is used to describe the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In this problem, we need to find all the angles whose cosine value is 0.5.

To solve this problem, we need to use the inverse cosine function, denoted by $\cos^{-1}x$, which is the angle whose cosine is equal to x. We can use a calculator or a trigonometric table to find the angle whose cosine is 0.5.

Using a calculator, we find that $\cos^{-1}(0.5) = 60^{\circ}$ and $\cos^{-1}(0.5) = 120^{\circ}$.

Problem #28: Finding the Angles with a Sine Value of 0.8

The sine function is used to describe the ratio of the opposite side to the hypotenuse in a right-angled triangle. In this problem, we need to find all the angles whose sine value is 0.8.

To solve this problem, we need to use the inverse sine function, denoted by $\sin^{-1}x$, which is the angle whose sine is equal to x. We can use a calculator or a trigonometric table to find the angle whose sine is 0.8.

Using a calculator, we find that $\sin^{-1}(0.8) = 53.13^{\circ}$ and $\sin^{-1}(0.8) = 126.87^{\circ}$.

Problem #29: Finding the Angles with a Tangent Value of 2

The tangent function is used to describe the ratio of the opposite side to the adjacent side in a right-angled triangle. In this problem, we need to find all the angles whose tangent value is 2.

To solve this problem, we need to use the inverse tangent function, denoted by $\tan^{-1}x$, which is the angle whose tangent is equal to x. We can use a calculator or a trigonometric table to find the angle whose tangent is 2.

Using a calculator, we find that $\tan^{-1}(2) = 63.43^{\circ}$ and $\tan^{-1}(2) = 116.57^{\circ}$.

Conclusion

In this article, we have discussed various problems related to trigonometric functions, specifically the inverse trigonometric functions. We have used the inverse sine, cosine, and tangent functions to find the angles whose sine, cosine, and tangent values are given. These problems are essential in mathematics and are used in various scientific and engineering applications.
Frequently Asked Questions (FAQs) on Trigonometric Functions

Q: What is the difference between trigonometric functions and inverse trigonometric functions?

A: Trigonometric functions are used to describe the relationships between the sides and angles of triangles, while inverse trigonometric functions are used to find the angles whose trigonometric values are given.

Q: What are the six basic trigonometric functions?

A: The six basic trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant.

Q: What is the unit circle and how is it used in trigonometry?

A: The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is used to define the values of trigonometric functions for common angles.

Q: How do I use a calculator to find the values of trigonometric functions?

A: To use a calculator to find the values of trigonometric functions, you need to enter the angle in degrees or radians and select the trigonometric function you want to evaluate.

Q: What is the difference between degrees and radians?

A: Degrees are a unit of angle measurement, while radians are a unit of angle measurement that is based on the ratio of the arc length to the radius of a circle.

Q: How do I convert between degrees and radians?

A: To convert between degrees and radians, you can use the following formulas:

  • Degrees to radians: $\text{radians} = \frac{\pi}{180} \times \text{degrees}$
  • Radians to degrees: $\text{degrees} = \frac{180}{\pi} \times \text{radians}$

Q: What is the identity of the sine function?

A: The identity of the sine function is $\sin^2(x) + \cos^2(x) = 1$.

Q: What is the identity of the cosine function?

A: The identity of the cosine function is $\cos^2(x) + \sin^2(x) = 1$.

Q: What is the identity of the tangent function?

A: The identity of the tangent function is $\tan(x) = \frac{\sin(x)}{\cos(x)}$.

Q: How do I use the Pythagorean identity to find the values of trigonometric functions?

A: The Pythagorean identity can be used to find the values of trigonometric functions by rearranging the equation to isolate the desired function.

Q: What is the difference between the sine and cosine functions?

A: The sine function is used to describe the ratio of the opposite side to the hypotenuse in a right-angled triangle, while the cosine function is used to describe the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

Q: What is the difference between the tangent and cotangent functions?

A: The tangent function is used to describe the ratio of the opposite side to the adjacent side in a right-angled triangle, while the cotangent function is used to describe the ratio of the adjacent side to the opposite side in a right-angled triangle.

Q: How do I use the inverse trigonometric functions to find the angles whose trigonometric values are given?

A: To use the inverse trigonometric functions to find the angles whose trigonometric values are given, you need to enter the trigonometric value and select the inverse trigonometric function you want to evaluate.

Q: What are the common angles and their corresponding trigonometric values?

A: The common angles and their corresponding trigonometric values are:

  • 0 degrees: $\sin(0) = 0, \cos(0) = 1, \tan(0) = 0$
  • 30 degrees: $\sin(30) = \frac{1}{2}, \cos(30) = \frac{\sqrt{3}}{2}, \tan(30) = \frac{1}{\sqrt{3}}$
  • 45 degrees: $\sin(45) = \frac{\sqrt{2}}{2}, \cos(45) = \frac{\sqrt{2}}{2}, \tan(45) = 1$
  • 60 degrees: $\sin(60) = \frac{\sqrt{3}}{2}, \cos(60) = \frac{1}{2}, \tan(60) = \sqrt{3}$
  • 90 degrees: $\sin(90) = 1, \cos(90) = 0, \tan(90) = \infty$

Q: How do I use the trigonometric identities to simplify expressions involving trigonometric functions?

A: To use the trigonometric identities to simplify expressions involving trigonometric functions, you need to identify the relevant identity and apply it to the expression.

Q: What are the applications of trigonometry in real-life situations?

A: Trigonometry has numerous applications in real-life situations, including:

  • Navigation: Trigonometry is used in navigation to determine the position and direction of objects.
  • Physics: Trigonometry is used in physics to describe the motion of objects and the forces acting on them.
  • Engineering: Trigonometry is used in engineering to design and build structures such as bridges and buildings.
  • Computer Science: Trigonometry is used in computer science to develop algorithms and models for image and video processing.

Q: How do I use trigonometry to solve problems in mathematics and science?

A: To use trigonometry to solve problems in mathematics and science, you need to apply the relevant trigonometric functions and identities to the problem and use algebraic manipulations to simplify the expression and solve for the unknown variable.