Answer: To write an expression as a trigonometric function of one number using an addition or subtraction formula, we can use the trigonometric identities to rewrite the expression in terms of a single trigonometric function.
Let's consider the addition formula for sine:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
and the subtraction formula for sine:
sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
Similarly, we have the addition formula for cosine:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
and the subtraction formula for cosine:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
To use these formulas, let's take an example. Suppose we have the expression:
sin(30° + 45°)
Using the addition formula for sine, we can rewrite this expression as:
sin(30°)cos(45°) + cos(30°)sin(45°)
We can simplify further by substituting the values of sin(30°), cos(45°), cos(30°), and sin(45°) from the trigonometric unit circle or table. Let's assume sin(30°) = 1/2, cos(45°) = √2/2, cos(30°) = √3/2, and sin(45°) = √2/2.
Substituting these values, we get:
(1/2)(√2/2) + (√3/2)(√2/2)
Simplifying this expression, we have:
(√2/4) + (√6/4)
Combining like terms, we get:
(√2 + √6)/4
Therefore, sin(30° + 45°) can be written as (√2 + √6)/4.
In a similar manner, you can use the addition or subtraction formulas for sine and cosine to write expressions as a trigonometric function of one number. Just make sure to substitute the values of sine and cosine of the angles involved and simplify the expression using basic arithmetic operations.
Explanation: