Question

for each multiple choice or true/false question, choose the best answer. for each matching question, click on the dropdown box beside each choice and click on the number. for each short answer, type your responses in appropriate box numbers below the questions.

question 12 (1 point)
the six addition and subtraction compound angle formulas may be used to create all equivalent double angle formulas for sine, cosine and tangent.

true
false

Explanation:

⚡ Using what you learned: Trigonometric Identities

Step 1: Analyze the relationship between compound and double angle formulas

Double angle formulas are special cases of the compound angle (addition) formulas where the two angles are equal, i.e., \( A = B = \theta \).

Step 2: Derive the double angle formulas

Using the addition formulas:

• Sine:

$$ \sin(A + B) = \sin A \cos B + \cos A \sin B $$

Letting \( A = B = \theta \):

$$ \sin(2\theta) = \sin\theta \cos\theta + \cos\theta \sin\theta = 2\sin\theta \cos\theta $$

• Cosine:

$$ \cos(A + B) = \cos A \cos B - \sin A \sin B $$

Letting \( A = B = \theta \):

$$ \cos(2\theta) = \cos^2\theta - \sin^2\theta $$

Using the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \), we can also derive the other two equivalent forms:

$$ \cos(2\theta) = 2\cos^2\theta - 1 $$

$$ \cos(2\theta) = 1 - 2\sin^2\theta $$

• Tangent:

$$ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} $$

Letting \( A = B = \theta \):

$$ \tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta} $$

Since all double angle formulas for sine, cosine, and tangent can be derived directly from the addition formulas, the statement is true.

Answer:

True