Math Problem Statement

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

Solution

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(20α)\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) sinα4\sin\frac{\alpha}{4}:

    • There isn’t a specific sum, difference, or double angle formula directly related to sinα4\sin\frac{\alpha}{4}, so this might require additional context for further expansion.
  3. i) sin(4)cos(1)+cos(4)sin(1)\sin(4)\cos(1) + \cos(4)\sin(1):

    • This expression is of the form sinAcosB+cosAsinB\sin A \cos B + \cos A \sin B, which can be rewritten as sin(A+B)\sin(A + B).
    • Applying the formula: sin(4+1)=sin(5)\sin(4 + 1) = \sin(5).
  4. j) sin(πα)\sin(\pi - \alpha):

    • This is the sine of a difference: sin(πα)=sin(α)\sin(\pi - \alpha) = \sin(\alpha). This is a known identity from trigonometry.
  5. k) 2sin(6α)cos(6α)2\sin(6\alpha)\cos(6\alpha):

    • This is the sine double angle formula: 2sinAcosA=sin(2A)2\sin A \cos A = \sin(2A).
    • Applying this formula: 2sin(6α)cos(6α)=sin(12α)2\sin(6\alpha)\cos(6\alpha) = \sin(12\alpha).
  6. l) 2sinπ4cosπ42\sin\frac{\pi}{4}\cos\frac{\pi}{4}:

    • Again, using the sine double angle formula: 2sinAcosA=sin(2A)2\sin A \cos A = \sin(2A).
    • Here: 2sinπ4cosπ4=sin(2π4)=sinπ2=12\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(πα)\cos(\pi - \alpha)?
  3. Can you express sin(2A)\sin(2A) in terms of only sin(A)\sin(A) and cos(A)\cos(A)?
  4. What is the sum-to-product identity for sinA+sinB\sin A + \sin B?
  5. How would you solve cos(2α)\cos(2\alpha) using a double angle formula?

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

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Sine Sum and Difference
Double Angle Formulas

Formulas

sin(A ± B) = sinA cosB ± cosA sinB
2sinA cosA = sin(2A)
sin(π - α) = sin(α)

Theorems

Sum of angles for sine
Difference of angles for sine
Double angle formulas

Suitable Grade Level

Grades 9-12