Finding the Values of Trigonometric Functions

Oct 13, 2025 by pynkocean.com 69 views

Introduction

In this article, we will explore the relationship between a point on the unit circle and the values of trigonometric functions at a given real number $t$ . We will use the point $P\left(\frac{21}{29}, -\frac{20}{29}\right)$ on the unit circle to find the values of sine, cosine, and other trigonometric functions at $t$ .

The Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is defined by the equation $x^2 + y^2 = 1$ . The unit circle is a fundamental concept in trigonometry and is used to define the trigonometric functions.

The Point P

The point $P$ is given by the coordinates $\left(\frac{21}{29}, -\frac{20}{29}\right)$ . This point lies on the unit circle, and we can use it to find the values of the trigonometric functions at $t$ .

Finding the Values of Trigonometric Functions

To find the values of the trigonometric functions at $t$ , we need to use the coordinates of the point $P$ . We can use the following formulas to find the values of sine, cosine, and other trigonometric functions:

$\sin(t) = y$
$\cos(t) = x$
$\tan(t) = \frac{y}{x}$
$\csc(t) = \frac{1}{y}$
$\sec(t) = \frac{1}{x}$
$\cot(t) = \frac{x}{y}$

Finding the Value of Sine

Using the formula $\sin(t) = y$ , we can find the value of sine at $t$ :

\sin(t) = -\frac{20}{29}

Finding the Value of Cosine

Using the formula $\cos(t) = x$ , we can find the value of cosine at $t$ :

\cos(t) = \frac{21}{29}

Finding the Value of Tangent

Using the formula $\tan(t) = \frac{y}{x}$ , we can find the value of tangent at $t$ :

\tan(t) = \frac{-\frac{20}{29}}{\frac{21}{29}} = -\frac{20}{21}

Finding the Value of Cosecant

Using the formula $\csc(t) = \frac{1}{y}$ , we can find the value of cosecant at $t$ :

\csc(t) = \frac{1}{-\frac{20}{29}} = -\frac{29}{20}

Finding the Value of Secant

Using the formula $\sec(t) = \frac{1}{x}$ , we can find the value of secant at $t$ :

\sec(t) = \frac{1}{\frac{21}{29}} = \frac{29}{21}

Finding the Value of Cotangent

Using the formula $\cot(t) = \frac{x}{y}$ , we can find the value of cotangent at $t$ :

\cot(t) = \frac{\frac{21}{29}}{-\frac{20}{29}} = -\frac{21}{20}

Conclusion

In this article, we have used the point $P\left(\frac{21}{29}, -\frac{20}{29}\right)$ on the unit circle to find the values of the trigonometric functions at $t$ . We have used the formulas for sine, cosine, tangent, cosecant, secant, and cotangent to find the values of these functions. The values of these functions are:

$\sin(t) = -\frac{20}{29}$
$\cos(t) = \frac{21}{29}$
$\tan(t) = -\frac{20}{21}$
$\csc(t) = -\frac{29}{20}$
$\sec(t) = \frac{29}{21}$
$\cot(t) = -\frac{21}{20}$

These values can be used to solve problems involving the trigonometric functions and the unit circle.

References

[1] "Trigonometry" by Michael Corral, 2019.
[2] "Unit Circle" by Khan Academy, 2020.
[3] "Trigonometric Functions" by Math Open Reference, 2020.

Glossary

Unit Circle: A circle with a radius of 1 centered at the origin of the coordinate plane.
Trigonometric Functions: Functions that relate the angles of a triangle to the ratios of the lengths of its sides.
Sine: The ratio of the length of the side opposite an angle to the length of the hypotenuse.
Cosine: The ratio of the length of the side adjacent to an angle to the length of the hypotenuse.
Tangent: The ratio of the length of the side opposite an angle to the length of the side adjacent to an angle.
Cosecant: The reciprocal of the sine function.
Secant: The reciprocal of the cosine function.
Cotangent: The reciprocal of the tangent function.
A Point on the Unit Circle: Trigonometric Functions Q&A ===========================================================

Introduction

In our previous article, we explored the relationship between a point on the unit circle and the values of trigonometric functions at a given real number $t$ . We used the point $P\left(\frac{21}{29}, -\frac{20}{29}\right)$ on the unit circle to find the values of sine, cosine, and other trigonometric functions at $t$ . In this article, we will answer some frequently asked questions about the unit circle and trigonometric functions.

Q&A

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is defined by the equation $x^2 + y^2 = 1$ .

Q: What is the relationship between the unit circle and trigonometric functions?

A: The unit circle is used to define the trigonometric functions. The coordinates of a point on the unit circle can be used to find the values of sine, cosine, and other trigonometric functions.

Q: How do I find the values of trigonometric functions using the unit circle?

A: To find the values of trigonometric functions using the unit circle, you need to use the coordinates of a point on the unit circle. You can use the following formulas to find the values of sine, cosine, and other trigonometric functions:

$\sin(t) = y$
$\cos(t) = x$
$\tan(t) = \frac{y}{x}$
$\csc(t) = \frac{1}{y}$
$\sec(t) = \frac{1}{x}$
$\cot(t) = \frac{x}{y}$

Q: What is the value of sine at $t$ ?

A: The value of sine at $t$ is given by the formula $\sin(t) = y$ . Using the coordinates of the point $P\left(\frac{21}{29}, -\frac{20}{29}\right)$ , we can find the value of sine at $t$ :

\sin(t) = -\frac{20}{29}

Q: What is the value of cosine at $t$ ?

A: The value of cosine at $t$ is given by the formula $\cos(t) = x$ . Using the coordinates of the point $P\left(\frac{21}{29}, -\frac{20}{29}\right)$ , we can find the value of cosine at $t$ :

\cos(t) = \frac{21}{29}

Q: What is the value of tangent at $t$ ?

A: The value of tangent at $t$ is given by the formula $\tan(t) = \frac{y}{x}$ . Using the coordinates of the point $P\left(\frac{21}{29}, -\frac{20}{29}\right)$ , we can find the value of tangent at $t$ :

\tan(t) = \frac{-\frac{20}{29}}{\frac{21}{29}} = -\frac{20}{21}

Q: What is the value of cosecant at $t$ ?

A: The value of cosecant at $t$ is given by the formula $\csc(t) = \frac{1}{y}$ . Using the coordinates of the point $P\left(\frac{21}{29}, -\frac{20}{29}\right)$ , we can find the value of cosecant at $t$ :

\csc(t) = \frac{1}{-\frac{20}{29}} = -\frac{29}{20}

Q: What is the value of secant at $t$ ?

A: The value of secant at $t$ is given by the formula $\sec(t) = \frac{1}{x}$ . Using the coordinates of the point $P\left(\frac{21}{29}, -\frac{20}{29}\right)$ , we can find the value of secant at $t$ :

\sec(t) = \frac{1}{\frac{21}{29}} = \frac{29}{21}

Q: What is the value of cotangent at $t$ ?

A: The value of cotangent at $t$ is given by the formula $\cot(t) = \frac{x}{y}$ . Using the coordinates of the point $P\left(\frac{21}{29}, -\frac{20}{29}\right)$ , we can find the value of cotangent at $t$ :

\cot(t) = \frac{\frac{21}{29}}{-\frac{20}{29}} = -\frac{21}{20}

Conclusion

In this article, we have answered some frequently asked questions about the unit circle and trigonometric functions. We have used the point $P\left(\frac{21}{29}, -\frac{20}{29}\right)$ on the unit circle to find the values of sine, cosine, and other trigonometric functions at $t$ . We hope that this article has been helpful in understanding the relationship between the unit circle and trigonometric functions.

References

[1] "Trigonometry" by Michael Corral, 2019.
[2] "Unit Circle" by Khan Academy, 2020.
[3] "Trigonometric Functions" by Math Open Reference, 2020.

Glossary

Unit Circle: A circle with a radius of 1 centered at the origin of the coordinate plane.
Trigonometric Functions: Functions that relate the angles of a triangle to the ratios of the lengths of its sides.
Sine: The ratio of the length of the side opposite an angle to the length of the hypotenuse.
Cosine: The ratio of the length of the side adjacent to an angle to the length of the hypotenuse.
Tangent: The ratio of the length of the side opposite an angle to the length of the side adjacent to an angle.
Cosecant: The reciprocal of the sine function.
Secant: The reciprocal of the cosine function.
Cotangent: The reciprocal of the tangent function.