https://file.aiquickdraw.com/mathbot/file/1726674512691vmietrzl.jpg

The image contains a mathematical exercise with the instruction: *"Pritaikyk sumos arba skirtumo sinuso arba dvigubojo kampo sinuso formulę"* which translates to *"Apply the sine sum, difference, or double angle formulas for sine."* 

Let's break down each part of the task:

1. **g) \(\sin(20\alpha)\)**: 
   - No specific formula is given to expand this further. However, you could use the sine of multiple angles formula, but that may not be needed here.

2. **h) \(\sin\frac{\alpha}{4}\)**:
   - There isn’t a specific sum, difference, or double angle formula directly related to \(\sin\frac{\alpha}{4}\), so this might require additional context for further expansion.

3. **i) \(\sin(4)\cos(1) + \cos(4)\sin(1)\)**:
   - This expression is of the form \(\sin A \cos B + \cos A \sin B\), which can be rewritten as \(\sin(A + B)\).
   - Applying the formula: \(\sin(4 + 1) = \sin(5)\).

4. **j) \(\sin(\pi - \alpha)\)**:
   - This is the sine of a difference: \(\sin(\pi - \alpha) = \sin(\alpha)\). This is a known identity from trigonometry.

5. **k) \(2\sin(6\alpha)\cos(6\alpha)\)**:
   - This is the sine double angle formula: \(2\sin A \cos A = \sin(2A)\).
   - Applying this formula: \(2\sin(6\alpha)\cos(6\alpha) = \sin(12\alpha)\).

6. **l) \(2\sin\frac{\pi}{4}\cos\frac{\pi}{4}\)**:
   - Again, using the sine double angle formula: \(2\sin A \cos A = \sin(2A)\).
   - Here: \(2\sin\frac{\pi}{4}\cos\frac{\pi}{4} = \sin\left(2\cdot\frac{\pi}{4}\right) = \sin\frac{\pi}{2} = 1\).

Would you like more details on any specific step, or do you have further questions? Here are five questions to expand on the material:

1. How would you apply the double angle formula for cosine?
2. What is the identity for \(\cos(\pi - \alpha)\)?
3. Can you express \(\sin(2A)\) in terms of only \(\sin(A)\) and \(\cos(A)\)?
4. What is the sum-to-product identity for \(\sin A + \sin B\)?
5. How would you solve \(\cos(2\alpha)\) using a double angle formula?

**Tip**: Remember that for trigonometric identities, the sine of complementary angles like \(\sin(\pi - \alpha)\) can simplify expressions significantly!

Apply the sine sum, difference, or double angle formulas to solve the following trigonometric expressions.

Explore how to apply trigonometric identities such as the sine sum, difference, and double angle formulas to solve various expressions. This guide is suitable for high school students in Grades 9-12 and provides step-by-step solutions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation