How do I do this by sketching a triangle.
sin( arcsin(1/2) + arccos(1/2) )
Thanks.
let's start with arcsin(1/2)
draw a right angled triangle where theta and alpha are the two angles other than the 90º angle.
suppose you make the opposite side 1, and the hypotenuse 2.
Now look at the other angle alpha. Wouldn't the cosine of that angle also be 1/2??
(if you know your 30º, 60º, 90º triangle ratios, you will realize that the two angles are 30 and 60)
but since theta + alpha must total 90º, you are really just being asked for sin 90º, which is 1
so the answer is 1
To sketch the triangle and find the value of sin(arcsin(1/2) + arccos(1/2)), follow these steps:
1. Start with arcsin(1/2): In a right-angled triangle, let's label one acute angle as theta and the other as alpha. Set the opposite side as 1 and the hypotenuse as 2.
2. Find alpha: Since arcsin(1/2) = 30 degrees in this case (theta is already labeled as 90 degrees because it's the right angle), alpha must be 60 degrees because the sum of the angles in a triangle is 180 degrees.
3. Determine cos(alpha): In a 30-60-90 triangle, you can observe that the cosine of the 60-degree angle is also 1/2.
4. Combine the angles: Since theta + alpha = 90 degrees, you are essentially finding sin(90 degrees) = 1.
Therefore, the value of sin(arcsin(1/2) + arccos(1/2)) is 1.