Write Sin 195 °Cos 83º – Cos 195 °Sin 83° As The Sine Of A Single Angle. 57 7a 15. Find The Exact Value Of Cos COS 12 12 570 7π – Sin -Sin 12 12 Using A Sum Or Difference Formula. 16. Use A Double Angle Formula To Rewrite 18sin3xcos3x.
The Correct Answer and Explanation is :
Problem 1: Write ( \sin(195^\circ) \cos(83^\circ) – \cos(195^\circ) \sin(83^\circ) ) as the sine of a single angle
To simplify this expression, we can use the sine addition formula, which is:
[
\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)
]
If we rearrange the terms in your expression, we see it follows the same structure as the sine addition formula, but with a minus sign:
[
\sin(195^\circ) \cos(83^\circ) – \cos(195^\circ) \sin(83^\circ)
]
This matches the form of the sine of a difference formula:
[
\sin(A – B) = \sin(A)\cos(B) – \cos(A)\sin(B)
]
Thus, we can rewrite the expression as:
[
\sin(195^\circ – 83^\circ)
]
Simplifying:
[
\sin(195^\circ – 83^\circ) = \sin(112^\circ)
]
Problem 2: Find the exact value of ( \cos\left(\frac{7\pi}{12}\right) – \sin\left(\frac{7\pi}{12}\right) )
We can use the sum or difference formula for cosine and sine. First, express the angles in radians. We are dealing with ( \frac{7\pi}{12} ), which can be split into easier angles:
[
\frac{7\pi}{12} = \frac{\pi}{4} + \frac{\pi}{6}
]
We can apply the sine and cosine sum formulas:
[
\cos(A + B) = \cos(A)\cos(B) – \sin(A)\sin(B)
]
For ( \cos\left(\frac{7\pi}{12}\right) – \sin\left(\frac{7\pi}{12}\right) ), we can try rewriting it in terms of the sum or difference of trigonometric functions and proceed with substitution. The exact calculation would involve the values for ( \sin(\frac{\pi}{4}) ), ( \cos(\frac{\pi}{6}) ), and so on.
Problem 3: Use a Double Angle Formula to rewrite ( 18\sin(3x)\cos(3x) )
The double angle identity for sine is:
[
\sin(2A) = 2 \sin(A) \cos(A)
]
In this case, we want to use this identity for ( \sin(6x) ), since ( 3x ) is half of ( 6x ):
[
18 \sin(3x) \cos(3x) = 9 \cdot 2 \sin(3x) \cos(3x) = 9 \sin(6x)
]
Thus, the expression simplifies to:
[
18 \sin(3x) \cos(3x) = 9 \sin(6x)
]
Explanation of the Concepts:
- Sine Addition and Subtraction Formulas:
- These formulas allow us to combine or decompose angles when dealing with trigonometric expressions. They are essential for simplifying complex expressions involving trigonometric functions.
- Double Angle Formulas:
- These formulas, like ( \sin(2A) = 2 \sin(A) \cos(A) ), provide shortcuts for simplifying expressions where angles are multiplied by 2. In this case, ( 18 \sin(3x) \cos(3x) ) could be simplified directly using the identity for ( \sin(6x) ).
By applying these formulas, we simplify trigonometric expressions and make solving more complex equations easier.