Evaluate the expression cos75°sin15°+sin75°cos15°
Cos75 = sin15.
sin75 = Cos15.
Cos75 * Cos75 + sin75 * sin75 = Cos^2(75) + sin^2(75) = 1.
Ted/David/Chad -- why are you switching names?
recall that sin(A+B) = sinA cosB + cosA sinB
so, what do you think?
To evaluate the expression cos75°sin15°+sin75°cos15°, we can use the trigonometric identity formula for the sine and cosine of the sum of two angles.
The formula states that for any angles α and β, we have:
sin(α + β) = sinαcosβ + cosαsinβ
cos(α + β) = cosαcosβ - sinαsinβ
In our case, we have α = 75° and β = 15°. Let's substitute these values into the formulas to simplify the expression.
First, let's calculate sin(75° + 15°):
sin(75° + 15°) = sin75°cos15° + cos75°sin15°
Now, let's calculate cos(75° + 15°):
cos(75° + 15°) = cos75°cos15° - sin75°sin15°
We can see that the expression we want to evaluate, cos75°sin15° + sin75°cos15°, matches the sin(75° + 15°) formula. Therefore, we can directly use the trigonometric identity:
cos75°sin15° + sin75°cos15° = sin(75° + 15°)
Now, let's calculate sin(75° + 15°) using a trigonometric calculator or table:
sin(75° + 15°) ≈ sin90°
We know that sin90° equals 1. Therefore, the value of the expression cos75°sin15° + sin75°cos15° is approximately equal to 1.