https://r2.erweima.ai/mathbot/image/1726273164806cfbc5m4n.jpg

Let's solve the two parts of the problem.

### Part (a) - Evaluate the expression:
\[
\cos\left(\sin^{-1}\left(\frac{5}{11}\right)\right)
\]
Here, \(\sin^{-1}\left(\frac{5}{11}\right)\) gives an angle \(\theta\) such that \(\sin(\theta) = \frac{5}{11}\).

Using the Pythagorean identity for sine and cosine:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
We already know \(\sin(\theta) = \frac{5}{11}\), so:
\[
\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \left(\frac{5}{11}\right)^2 = 1 - \frac{25}{121} = \frac{96}{121}
\]
Thus, \(\cos(\theta) = \frac{\sqrt{96}}{11} = \frac{4\sqrt{6}}{11}\).

Therefore:
\[
\cos\left(\sin^{-1}\left(\frac{5}{11}\right)\right) = \frac{4\sqrt{6}}{11}
\]

### Part (b) - Solve the equation:
\[
e^{3 - 2x} = 6
\]
Take the natural logarithm of both sides:
\[
\ln\left(e^{3 - 2x}\right) = \ln(6)
\]
This simplifies to:
\[
3 - 2x = \ln(6)
\]
Solving for \(x\):
\[
-2x = \ln(6) - 3
\]
\[
x = \frac{3 - \ln(6)}{2}
\]
The value of \(\ln(6) \approx 1.7918\), so:
\[
x = \frac{3 - 1.7918}{2} = \frac{1.2082}{2} \approx 0.6041
\]

### Final Answers:
- (a) \(\cos\left(\sin^{-1}\left(\frac{5}{11}\right)\right) = \frac{4\sqrt{6}}{11}\)
- (b) \(x \approx 0.6041\)

Do you need more details or have any questions?

Here are five related questions:
1. How can you generalize the solution for \(\cos(\sin^{-1}(x))\)?
2. What other trigonometric identities can be used to simplify inverse functions?
3. Can the equation in part (b) be solved using other methods besides logarithms?
4. How does the derivative of \(e^x\) play a role in solving exponential equations?
5. How can the Pythagorean identity be applied to solve complex trigonometric problems?

**Tip:** When solving equations involving logarithms, remember that you can only take the logarithm of positive numbers.

Resuelva los siguientes ejercicios: a) Evalúe la expresión cos(sen^{-1}(5/11)), b) Resuelva la ecuación e^(3 - 2x) = 6

In this tutorial, we solve two math problems: evaluating cos(sin^(-1)(5/11)) using trigonometric identities and solving the exponential equation e^(3 - 2x) = 6 with logarithms. This detailed solution covers key trigonometric and algebraic concepts for students in grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation