Find the exact value of sin(1/2arcsin(-7/25))
suspect you mean
sin[ (1/2) arc sin -7/25) ]
draw your right triangle
call our angle A in quadrant 3 or 4 use 3
sides are -7 and -24, hypotenuse = 25
sinA = -7/25
cosA = -24/25
Half angle
sin (.5A) = - sqrt( [1-cosA]/2)
= -sqrt( [ 1/25 ] /2 )
= -(1/5)sqrt (1/2)
To find the exact value of sin(1/2arcsin(-7/25)), we need to break it down step by step.
Step 1: Let's start by finding the value of arcsin(-7/25). This involves finding an angle whose sine value is -7/25. Recall that arcsin returns an angle, not a ratio or decimal.
We have: arcsin(-7/25) = θ
Step 2: Now, we need to find the value of θ in terms of degrees or radians. To do this, we can use a calculator that has an arcsin function. Simply input -7/25 into the arcsin function to obtain the angle.
Assuming we are working in degrees, we find that θ ≈ -16.9°.
Step 3: Next, we need to find sin(1/2θ). This involves taking half of the angle obtained in Step 2 and determining the sine of that angle.
We have: sin(1/2θ) = sin(-8.45°)
Step 4: Now, we can find the value of sin(-8.45°) using a calculator that has a sine function. The sine of -8.45° is approximately -0.147.
Therefore, sin(1/2arcsin(-7/25)) ≈ -0.147.