Preview 5 out of 10 Flashcards
### Question:
Using the double angle formula for sine, calculate (sin 2A) if (sin A = frac{3}{5}) and (A) is in the first quadrant.
### Question:
Using the double angle formula for sine, calculate (sin 2A) if (sin A = fr...
Given (sin A = frac{3}{5}) and knowing (A) is in the first quadrant, we can find (cos A) using the Pythagorean identity.

1. First, find (cos A):
 [
 sin^2 A + cos^2 A = 1
 ]
 [
 left(frac{3}{5}right)^2 + cos^2 A = 1
 ]
 [
 frac{9}{25} + cos^2 A = 1
 ]
 [
 cos^2 A = 1 - frac{9}{25}
 ]
 [
 cos^2 A = frac{25}{25} - frac{9}{25}
 ]
 [
 cos^2 A = frac{16}{25}
 ]
 [
 cos A = sqrt{frac{16}{25}} = frac{4}{5}
 ]
 Since (A) is in the first quadrant, (cos A) is positive, so (cos A = frac{4}{5}).

2. Now, use the double angle formula for sine:
 [
 sin 2A = 2 sin A cos A
 ]
 Substitute the values:
 [
 sin 2A = 2 left(frac{3}{5}right) left(frac{4}{5}right)
 ]
 [
 sin 2A = 2 left(frac{12}{25}right)
 ]
 [
 sin 2A = frac{24}{25}
 ]

Thus, (sin 2A = frac{24}{25}).
Simplify (sec^2 x - tan^2 x).
Simplify (sec^2 x - tan^2 x).
Using the Pythagorean identity:
 [sec^2 x - tan^2 x = 1]
Prove that (tan(A + B) = frac{tan A + tan B}{1 - tan A tan B}).
Prove that (tan(A + B) = frac{tan A + tan B}{1 - tan A tan B}).
Using the angle addition formula for tangent, we have:
 [tan(A + B) = frac{tan A + tan B}{1 - tan A tan B}]
 This follows from the standard tangent addition formula.
Simplify (sin 2x).
Simplify (sin 2x).
Using the double-angle formula for sine:
 [sin 2x = 2 sin x cos x]
If (sin A = frac{5}{13}) and (A) is an acute angle, find (cos A).
If (sin A = frac{5}{13}) and (A) is an acute angle, find (cos A).
Since (A) is acute, (sin A = frac{5}{13}) implies:
 [sin^2 A + cos^2 A = 1]
 [left(frac{5}{13}right)^2 + cos^2 A = 1]
 [frac{25}{169} + cos^2 A = 1]
 [cos^2 A = 1 - frac{25}{169}]
 [cos^2 A = frac{169}{169} - frac{25}{169}]
 [cos^2 A = frac{144}{169}]
 [cos A = pm frac{12}{13}]

 Since (A) is an acute angle, (cos A) must be positive:
 [cos A = frac{12}{13}]