Exact Values of Trigonometric Functions for Common Angles

Oct 13, 2025 by windailys.com 66 views

Trigonometric Functions: Exact Values and Identities

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 explore the exact values of trigonometric functions for a given angle $\theta$ . We will also discuss the reciprocal identities and the Pythagorean identities that are essential in trigonometry.

Reciprocal Identities

The reciprocal identities are a set of relationships between the trigonometric functions. They are:

$\csc \theta = \frac{1}{\sin \theta}$
$\sec \theta = \frac{1}{\cos \theta}$
$\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$

These identities can be used to express the reciprocal functions in terms of the other trigonometric functions.

Pythagorean Identities

The Pythagorean identities are a set of relationships between the trigonometric functions that are based on the Pythagorean theorem. They are:

$\sin^2 \theta + \cos^2 \theta = 1$
$1 + \tan^2 \theta = \sec^2 \theta$
$1 + \cot^2 \theta = \csc^2 \theta$

These identities can be used to express the trigonometric functions in terms of each other.

Exact Values of Trigonometric Functions

The exact values of trigonometric functions for a given angle $\theta$ can be found using the reciprocal identities and the Pythagorean identities. Here are the exact values of the trigonometric functions for a given angle $\theta$ :

Sine, Cosine, and Tangent

$\sin \theta = \frac{1}{\csc \theta}$
$\cos \theta = \frac{1}{\sec \theta}$
$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{1}{\cot \theta}$

Cosecant, Secant, and Cotangent

$\csc \theta = \frac{1}{\sin \theta}$
$\sec \theta = \frac{1}{\cos \theta}$
$\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$

Pythagorean Identities

$\sin^2 \theta + \cos^2 \theta = 1$
$1 + \tan^2 \theta = \sec^2 \theta$
$1 + \cot^2 \theta = \csc^2 \theta$

Exact Values of Trigonometric Functions for Common Angles

Here are the exact values of the trigonometric functions for common angles:

$\sin 0^\circ = 0$
$\cos 0^\circ = 1$
$\tan 0^\circ = 0$
$\sin 30^\circ = \frac{1}{2}$
$\cos 30^\circ = \frac{\sqrt{3}}{2}$
$\tan 30^\circ = \frac{1}{\sqrt{3}}$
$\sin 45^\circ = \frac{1}{\sqrt{2}}$
$\cos 45^\circ = \frac{1}{\sqrt{2}}$
$\tan 45^\circ = 1$
$\sin 60^\circ = \frac{\sqrt{3}}{2}$
$\cos 60^\circ = \frac{1}{2}$
$\tan 60^\circ = \sqrt{3}$
$\sin 90^\circ = 1$
$\cos 90^\circ = 0$
$\tan 90^\circ = \infty$

Using the Reciprocal Identities

The reciprocal identities can be used to express the reciprocal functions in terms of the other trigonometric functions. For example:

$\csc \theta = \frac{1}{\sin \theta} = \frac{\sin \theta}{\sin^2 \theta}$
$\sec \theta = \frac{1}{\cos \theta} = \frac{\cos \theta}{\cos^2 \theta}$
$\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$

Using the Pythagorean Identities

The Pythagorean identities can be used to express the trigonometric functions in terms of each other. For example:

$\sin^2 \theta + \cos^2 \theta = 1$
$1 + \tan^2 \theta = \sec^2 \theta$
$1 + \cot^2 \theta = \csc^2 \theta$

Solving Trigonometric Equations

Trigonometric equations can be solved using the reciprocal identities and the Pythagorean identities. For example:

$\sin \theta = \frac{1}{2}$
$\cos \theta = \frac{\sqrt{3}}{2}$
$\tan \theta = \sqrt{3}$

Conclusion

In this article, we have discussed the exact values of trigonometric functions for a given angle $\theta$ . We have also discussed the reciprocal identities and the Pythagorean identities that are essential in trigonometry. The reciprocal identities can be used to express the reciprocal functions in terms of the other trigonometric functions, while the Pythagorean identities can be used to express the trigonometric functions in terms of each other. We have also provided examples of how to use the reciprocal identities and the Pythagorean identities to solve trigonometric equations.

References

"Trigonometry" by Michael Corral
"Trigonometry" by I. M. Gelfand and M. L. Gelfand
"Trigonometry" by Charles P. McKeague and Mark D. Turner

Further Reading

"Trigonometry: A Unit Circle Approach" by Michael Sullivan
"Trigonometry: A Graphing Approach" by Charles P. McKeague and Mark D. Turner
"Trigonometry: A Problem-Solving Approach" by Charles P. McKeague and Mark D. Turner
Trigonometry Q&A: Frequently Asked 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 of the most frequently asked questions about trigonometry.

Q: What is trigonometry?

A: 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.

Q: What are the basic trigonometric functions?

A: The basic trigonometric functions are:

Sine (sin)
Cosine (cos)
Tangent (tan)
Cosecant (csc)
Secant (sec)
Cotangent (cot)

Q: What is the difference between sine and cosine?

A: Sine and cosine are two basic trigonometric functions that are related to each other. Sine is the ratio of the length of the side opposite the angle to the length of the hypotenuse, while cosine is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in trigonometry that states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.

Q: How do I use the Pythagorean theorem to find the length of the hypotenuse?

A: To use the Pythagorean theorem to find the length of the hypotenuse, you need to know the lengths of the other two sides. You can then use the formula:

c^2 = a^2 + b^2

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

Q: What is the difference between tangent and cotangent?

A: Tangent and cotangent are two basic trigonometric functions that are related to each other. Tangent is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle, while cotangent is the ratio of the length of the side adjacent to the angle to the length of the side opposite the angle.

Q: How do I use the reciprocal identities to find the value of a trigonometric function?

A: To use the reciprocal identities to find the value of a trigonometric function, you need to know the value of the reciprocal function. For example, if you know the value of the sine function, you can use the reciprocal identity to find the value of the cosecant function.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 that is centered at the origin of a coordinate plane. It is a fundamental concept in trigonometry that is used to define the trigonometric functions.

Q: How do I use the unit circle to find the value of a trigonometric function?

A: To use the unit circle to find the value of a trigonometric function, you need to know the coordinates of a point on the unit circle. You can then use the coordinates to find the value of the trigonometric function.

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

A: A right triangle is a triangle with one right angle, while an oblique triangle is a triangle with no right angles.

Q: How do I use the law of sines to find the length of a side of a triangle?

A: To use the law of sines to find the length of a side of a triangle, you need to know the lengths of the other two sides and the measure of the angle opposite the side you want to find. You can then use the formula:

a / sin(A) = b / sin(B) = c / sin(C)

where a, b, and c are the lengths of the sides, and A, B, and C are the measures of the angles.

Q: What is the difference between the law of sines and the law of cosines?

A: The law of sines and the law of cosines are two fundamental concepts in trigonometry that are used to find the lengths of the sides of a triangle. The law of sines is used to find the length of a side when you know the lengths of the other two sides and the measure of the angle opposite the side you want to find. The law of cosines is used to find the length of a side when you know the lengths of the other two sides and the measure of the angle between them.

Q: How do I use the law of cosines to find the length of a side of a triangle?

A: To use the law of cosines to find the length of a side of a triangle, you need to know the lengths of the other two sides and the measure of the angle between them. You can then use the formula:

c^2 = a^2 + b^2 - 2ab * cos(C)

where c is the length of the side you want to find, and a and b are the lengths of the other two sides.

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

A: The sine and cosine functions are two basic trigonometric functions that are related to each other. The sine function is the ratio of the length of the side opposite the angle to the length of the hypotenuse, while the cosine function is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Q: How do I use the sine and cosine functions to find the length of a side of a triangle?

A: To use the sine and cosine functions to find the length of a side of a triangle, you need to know the lengths of the other two sides and the measure of the angle opposite the side you want to find. You can then use the formulas:

a = b * sin(A) / sin(B) b = a * sin(B) / sin(A)

where a and b are the lengths of the sides, and A and B are the measures of the angles.

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

A: The tangent and cotangent functions are two basic trigonometric functions that are related to each other. The tangent function is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle, while the cotangent function is the ratio of the length of the side adjacent to the angle to the length of the side opposite the angle.

Q: How do I use the tangent and cotangent functions to find the length of a side of a triangle?

A: To use the tangent and cotangent functions to find the length of a side of a triangle, you need to know the lengths of the other two sides and the measure of the angle opposite the side you want to find. You can then use the formulas:

a = b * tan(A) / tan(B) b = a * tan(B) / tan(A)

where a and b are the lengths of the sides, and A and B are the measures of the angles.

Q: What is the difference between the law of sines and the law of cosines?

Q: How do I use the law of sines and the law of cosines to find the length of a side of a triangle?

A: To use the law of sines and the law of cosines to find the length of a side of a triangle, you need to know the lengths of the other two sides and the measure of the angle opposite the side you want to find. You can then use the formulas:

a / sin(A) = b / sin(B) = c / sin(C) c^2 = a^2 + b^2 - 2ab * cos(C)

where a, b, and c are the lengths of the sides, and A, B, and C are the measures of the angles.

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

Q: How do I use the sine and cosine functions to find the length of a side of a triangle?

a = b * sin(A) / sin(B) b = a * sin(B) / sin(A)

where a and b are the lengths of the sides, and A and B are the measures of the angles.

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