A Point $P(x, Y$\] Is Shown On The Unit Circle Corresponding To A Real Number $t$. Find The Values Of The Trigonometric Functions At $t$.The Point $P$ Is $P\left(\frac{21}{29}, -\frac{20}{29}\right$\].a.

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.