Question

Sina=4/5; 90 <a <180; sin2a-? sin(a/2)-?

Answer 1

Sina=4/5;90°<a<180°
180°<2a<360°
cos²a=1-sin²a=1-16/25=(25-16)/25=9/25
cosa=-√(9/25)=-3/5
sin2a=2•sina•cosa=2•4/5•(-3/5)==-24/25
sin(a/2)=?
sin²(a/2)=(1-cosa)/2=(1+3/5)/2=8/10=4/5
90°<a<180°
45°<a/2<90°
sin(a/2)=+√(4/5)=2/√5=2√5/5

Answer 2

To find the values of sin(2a) and sin(a/2) given that sina = 4/5 and 90° < a < 180°, we'll first use the double-angle and half-angle identities for sine.

1. Double-angle identity for sine: sin(2a) = 2 * sin(a) * cos(a)

2. Half-angle identity for sine: sin(a/2) = ±√((1 - cos(a)) / 2)

Now, let's calculate the values step by step:

Step 1: Find cos(a) Since sina = 4/5, we can use the Pythagorean identity to find cos(a). cos(a) = √(1 - sin^2(a)) cos(a) = √(1 - (4/5)^2) cos(a) = √(1 - 16/25) cos(a) = √(9/25) cos(a) = 3/5

Step 2: Find sin(2a) sin(2a) = 2 * sin(a) * cos(a) sin(2a) = 2 * (4/5) * (3/5) sin(2a) = 24/25

Step 3: Find sin(a/2) sin(a/2) = ±√((1 - cos(a)) / 2) sin(a/2) = ±√((1 - 3/5) / 2) sin(a/2) = ±√((2/5) / 2) sin(a/2) = ±√(1/5) sin(a/2) = ±√5 / 5

Note: Since 90° < a < 180°, a is in the second quadrant where sine is positive and cosine is negative. Therefore, sin(a/2) is positive.

Final results: sin(2a) = 24/25 sin(a/2) = √5 / 5 (approximately 0.447)