Given: $ \sin (u) = \frac{3}{5} }$and { \cos (u)$}$ Is Negative.Find The Following $[ \begin{array {l} \cos (u) = \ \tan (u) = \ \sin (-u) = \ \cos (-u) = \ \tan (-u) = \ \sin (u + \pi) = \ \cos (u + \pi) = \ \tan (u +

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 explore how to solve trigonometric functions when given certain values. We will use the given values of $\sin (u) = \frac{3}{5}$ and $\cos (u)$ is negative to find the values of various trigonometric functions.

Finding Cosine, Tangent, and Sine Functions

Finding Cosine Function

To find the value of $\cos (u)$, we can use the Pythagorean identity: $\sin^2 (u) + \cos^2 (u) = 1$. Since we are given that $\sin (u) = \frac{3}{5}$, we can substitute this value into the equation:

$$\left(\frac{3}{5}\right)^2 + \cos^2 (u) = 1$$

Simplifying the equation, we get:

$$\frac{9}{25} + \cos^2 (u) = 1$$

Subtracting $\frac{9}{25}$ from both sides, we get:

$$\cos^2 (u) = \frac{16}{25}$$

Since $\cos (u)$ is negative, we can take the negative square root of both sides:

$$\cos (u) = -\frac{4}{5}$$

Finding Tangent Function

To find the value of $\tan (u)$, we can use the definition of tangent: $\tan (u) = \frac{\sin (u)}{\cos (u)}$. Substituting the given values, we get:

$$\tan (u) = \frac{\frac{3}{5}}{-\frac{4}{5}}$$

Simplifying the expression, we get:

$$\tan (u) = -\frac{3}{4}$$

Finding Sine Function for Negative Angle

To find the value of $\sin (-u)$, we can use the property of sine function: $\sin (-u) = -\sin (u)$. Substituting the given value, we get:

$$\sin (-u) = -\frac{3}{5}$$

Finding Cosine Function for Negative Angle

To find the value of $\cos (-u)$, we can use the property of cosine function: $\cos (-u) = \cos (u)$. Since we already found that $\cos (u) = -\frac{4}{5}$, we can substitute this value:

$$\cos (-u) = -\frac{4}{5}$$

Finding Tangent Function for Negative Angle

To find the value of $\tan (-u)$, we can use the definition of tangent: $\tan (-u) = \frac{\sin (-u)}{\cos (-u)}$. Substituting the values we found, we get:

$$\tan (-u) = \frac{-\frac{3}{5}}{-\frac{4}{5}}$$

Simplifying the expression, we get:

$$\tan (-u) = \frac{3}{4}$$

Finding Sine and Cosine Functions for a Sum of Angles

Finding Sine Function for a Sum of Angles

To find the value of $\sin (u + \pi)$, we can use the property of sine function: $\sin (u + \pi) = -\sin (u)$. Substituting the given value, we get:

$$\sin (u + \pi) = -\frac{3}{5}$$

Finding Cosine Function for a Sum of Angles

To find the value of $\cos (u + \pi)$, we can use the property of cosine function: $\cos (u + \pi) = -\cos (u)$. Since we already found that $\cos (u) = -\frac{4}{5}$, we can substitute this value:

$$\cos (u + \pi) = -\left(-\frac{4}{5}\right)$$

Simplifying the expression, we get:

$$\cos (u + \pi) = \frac{4}{5}$$

Finding Tangent Function for a Sum of Angles

To find the value of $\tan (u + \pi)$, we can use the definition of tangent: $\tan (u + \pi) = \frac{\sin (u + \pi)}{\cos (u + \pi)}$. Substituting the values we found, we get:

$$\tan (u + \pi) = \frac{-\frac{3}{5}}{\frac{4}{5}}$$

Simplifying the expression, we get:

$$\tan (u + \pi) = -\frac{3}{4}$$

Conclusion

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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 and provide solutions to various trigonometric problems.

Q: What is the difference between sine, cosine, and tangent?

A: Sine, cosine, and tangent are three fundamental trigonometric functions that are used to describe the relationships between the sides and angles of triangles. 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 find the value of sine, cosine, and tangent for a given angle?

A: To find the value of sine, cosine, and tangent for a given angle, you can use the unit circle or a trigonometric table. The unit circle is a circle with a radius of 1 that is centered at the origin of a coordinate plane. The sine, cosine, and tangent of an angle are equal to the ratios of the lengths of the sides of the triangle formed by the angle and the unit circle.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental equation in trigonometry that states that the sum of the squares of the sine and cosine of an angle is equal to 1. Mathematically, it can be expressed as:

$$\sin^2 (u) + \cos^2 (u) = 1$$

Q: How do I find the value of cosine for a given angle?

A: To find the value of cosine for a given angle, you can use the Pythagorean identity. If you know the value of sine, you can substitute it into the equation and solve for cosine.

Q: What is the difference between the sine and cosine of a negative angle?

A: The sine and cosine of a negative angle are equal to the negative of the sine and cosine of the positive angle. Mathematically, it can be expressed as:

$$\sin (-u) = -\sin (u)$$

$$\cos (-u) = \cos (u)$$

Q: How do I find the value of tangent for a given angle?

A: To find the value of tangent for a given angle, you can use the definition of tangent: $\tan (u) = \frac{\sin (u)}{\cos (u)}$. If you know the values of sine and cosine, you can substitute them into the equation and solve for tangent.

Q: What is the difference between the tangent of a positive and negative angle?

A: The tangent of a negative angle is equal to the negative of the tangent of the positive angle. Mathematically, it can be expressed as:

$$\tan (-u) = -\tan (u)$$

Q: How do I find the value of sine and cosine for a sum of angles?

A: To find the value of sine and cosine for a sum of angles, you can use the following formulas:

$$\sin (u + v) = \sin (u) \cos (v) + \cos (u) \sin (v)$$

$$\cos (u + v) = \cos (u) \cos (v) - \sin (u) \sin (v)$$

Q: What is the difference between the sine and cosine of a sum of angles?

A: The sine and cosine of a sum of angles are equal to the sum of the products of the sine and cosine of the individual angles. Mathematically, it can be expressed as:

$$\sin (u + v) = \sin (u) \cos (v) + \cos (u) \sin (v)$$

$$\cos (u + v) = \cos (u) \cos (v) - \sin (u) \sin (v)$$

Conclusion

In this article, we answered some frequently asked questions about trigonometry and provided solutions to various trigonometric problems. We hope that this article has been helpful in understanding the concepts of trigonometry and solving trigonometric problems.