In Exercises 57-64, Find The Exact Value Of The Following Under The Given Conditions:a. Cos ⁡ ( Α + Β \cos (\alpha+\beta Cos ( Α + Β ]b. Sin ⁡ ( Α + Β \sin (\alpha+\beta Sin ( Α + Β ]57. Sin ⁡ Α = 3 5 \sin \alpha = \frac{3}{5} Sin Α = 5 3 ​ , Α \alpha Α Lies In Quadrant I, And $\sin \beta =

In Exercises 57-64: Finding Exact Values of Trigonometric Expressions

In this article, we will delve into the world of trigonometry and explore the process of finding exact values of trigonometric expressions. Specifically, we will focus on exercises 57-64, where we are given the values of sine and cosine functions for two angles, $\alpha$ and $\beta$, and asked to find the exact values of $\cos (\alpha+\beta)$ and $\sin (\alpha+\beta)$. We will use the given conditions to determine the exact values of these trigonometric expressions.

Exercise 57: Finding $\cos (\alpha+\beta)$

Given Information

• $\sin \alpha = \frac{3}{5}$
• $\alpha$ lies in quadrant I
• $\sin \beta = \frac{4}{5}$

Finding $\cos \alpha$

Since $\alpha$ lies in quadrant I, we know that $\cos \alpha$ is positive. We can use the Pythagorean identity to find the value of $\cos \alpha$:

$$\cos^2 \alpha + \sin^2 \alpha = 1$$

Substituting the given value of $\sin \alpha$, we get:

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

Simplifying, we get:

$$\cos^2 \alpha + \frac{9}{25} = 1$$

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

$$\cos^2 \alpha = \frac{16}{25}$$

Taking the square root of both sides, we get:

$$\cos \alpha = \pm \frac{4}{5}$$

Since $\alpha$ lies in quadrant I, we know that $\cos \alpha$ is positive. Therefore, we have:

$$\cos \alpha = \frac{4}{5}$$

Finding $\cos \beta$

We are given that $\sin \beta = \frac{4}{5}$. Since $\beta$ is not specified to lie in a particular quadrant, we cannot determine the sign of $\cos \beta$. However, we can use the Pythagorean identity to find the value of $\cos \beta$:

$$\cos^2 \beta + \sin^2 \beta = 1$$

Substituting the given value of $\sin \beta$, we get:

$$\cos^2 \beta + \left(\frac{4}{5}\right)^2 = 1$$

Simplifying, we get:

$$\cos^2 \beta + \frac{16}{25} = 1$$

Subtracting $\frac{16}{25}$ from both sides, we get:

$$\cos^2 \beta = \frac{9}{25}$$

Taking the square root of both sides, we get:

$$\cos \beta = \pm \frac{3}{5}$$

Finding $\cos (\alpha+\beta)$

We can use the angle addition formula for cosine to find the value of $\cos (\alpha+\beta)$:

$$\cos (\alpha+\beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$$

Substituting the values we found earlier, we get:

$$\cos (\alpha+\beta) = \left(\frac{4}{5}\right)\left(\frac{3}{5}\right) - \left(\frac{3}{5}\right)\left(\frac{4}{5}\right)$$

Simplifying, we get:

$$\cos (\alpha+\beta) = \frac{12}{25} - \frac{12}{25}$$

Subtracting, we get:

$$\cos (\alpha+\beta) = 0$$

Therefore, the exact value of $\cos (\alpha+\beta)$ is 0.

Exercise 57: Finding $\sin (\alpha+\beta)$

Finding $\sin (\alpha+\beta)$

We can use the angle addition formula for sine to find the value of $\sin (\alpha+\beta)$:

$$\sin (\alpha+\beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$

Substituting the values we found earlier, we get:

$$\sin (\alpha+\beta) = \left(\frac{3}{5}\right)\left(\frac{3}{5}\right) + \left(\frac{4}{5}\right)\left(\frac{4}{5}\right)$$

Simplifying, we get:

$$\sin (\alpha+\beta) = \frac{9}{25} + \frac{16}{25}$$

Adding, we get:

$$\sin (\alpha+\beta) = \frac{25}{25}$$

Simplifying, we get:

$$\sin (\alpha+\beta) = 1$$

Therefore, the exact value of $\sin (\alpha+\beta)$ is 1.

In this article, we used the given conditions to find the exact values of $\cos (\alpha+\beta)$ and $\sin (\alpha+\beta)$. We found that $\cos (\alpha+\beta) = 0$ and $\sin (\alpha+\beta) = 1$. These results demonstrate the importance of using trigonometric identities and formulas to solve problems in mathematics. By applying these concepts, we can find exact values of trigonometric expressions and solve a wide range of mathematical problems.

• [1] "Trigonometry" by Michael Corral
• [2] "Precalculus" by James Stewart

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Trigonometry

Q: What is the value of $\cos (\alpha+\beta)$ when $\sin \alpha = \frac{3}{5}$, $\alpha$ lies in quadrant I, and $\sin \beta = \frac{4}{5}$?

A: The value of $\cos (\alpha+\beta)$ is 0.

Q: How do I find the value of $\cos \alpha$ when $\sin \alpha = \frac{3}{5}$ and $\alpha$ lies in quadrant I?

A: You can use the Pythagorean identity to find the value of $\cos \alpha$. The Pythagorean identity states that $\cos^2 \alpha + \sin^2 \alpha = 1$. Substituting the given value of $\sin \alpha$, you get $\cos^2 \alpha + \left(\frac{3}{5}\right)^2 = 1$. Simplifying, you get $\cos^2 \alpha = \frac{16}{25}$. Taking the square root of both sides, you get $\cos \alpha = \pm \frac{4}{5}$. Since $\alpha$ lies in quadrant I, you know that $\cos \alpha$ is positive. Therefore, you have $\cos \alpha = \frac{4}{5}$.

Q: How do I find the value of $\cos \beta$ when $\sin \beta = \frac{4}{5}$?

A: You can use the Pythagorean identity to find the value of $\cos \beta$. The Pythagorean identity states that $\cos^2 \beta + \sin^2 \beta = 1$. Substituting the given value of $\sin \beta$, you get $\cos^2 \beta + \left(\frac{4}{5}\right)^2 = 1$. Simplifying, you get $\cos^2 \beta = \frac{9}{25}$. Taking the square root of both sides, you get $\cos \beta = \pm \frac{3}{5}$.

Q: What is the value of $\sin (\alpha+\beta)$ when $\sin \alpha = \frac{3}{5}$, $\alpha$ lies in quadrant I, and $\sin \beta = \frac{4}{5}$?

A: The value of $\sin (\alpha+\beta)$ is 1.

Q: How do I use the angle addition formula for sine to find the value of $\sin (\alpha+\beta)$?

A: The angle addition formula for sine states that $\sin (\alpha+\beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$. Substituting the values you found earlier, you get $\sin (\alpha+\beta) = \left(\frac{3}{5}\right)\left(\frac{3}{5}\right) + \left(\frac{4}{5}\right)\left(\frac{4}{5}\right)$. Simplifying, you get $\sin (\alpha+\beta) = \frac{9}{25} + \frac{16}{25}$. Adding, you get $\sin (\alpha+\beta) = \frac{25}{25}$. Simplifying, you get $\sin (\alpha+\beta) = 1$.

Q: What are some common trigonometric identities that I can use to solve problems like this?

A: Some common trigonometric identities that you can use to solve problems like this include the Pythagorean identity, the angle addition formula for sine, and the angle addition formula for cosine.

Q: How can I apply these concepts to solve real-world problems?

A: You can apply these concepts to solve real-world problems by using trigonometry to model and analyze the behavior of waves, vibrations, and other periodic phenomena. You can also use trigonometry to solve problems in fields such as physics, engineering, and computer science.

In this article, we answered some frequently asked questions about trigonometry exercises 57-64. We covered topics such as finding the value of $\cos (\alpha+\beta)$, finding the value of $\cos \alpha$, finding the value of $\cos \beta$, and using the angle addition formula for sine to find the value of $\sin (\alpha+\beta)$. We also discussed some common trigonometric identities and how to apply these concepts to solve real-world problems.