find the exact values of s in the given intervals that have the given circular function values.
1. [pi/2,pi]; sin s=square root 2/2
2. [pi/2,pi]; cos s=-square root 3/2
these are simple angles
sin 45 degrees = (sqrt 2) /2
cos 30 deg = (sqrt 3)/2
But
Look, you need a calculator. Here is one.
For example for the angle whose sin is .75
type asin(.75) and click on the equal sign.
http://www.mathsisfun.com/scientific-calculator.html
Oh, did not notice between 90 and 180 degrees
180 - 45 = 135 degrees
180 - 30 = 150 degrees
To find the exact values of s in the given intervals that have the given circular function values, we will use the unit circle and solve for s.
1. [π/2, π]; sin(s) = √2/2
On the unit circle, the point at angle s has coordinates (cos(s), sin(s)). Since sin(s) = √2/2, we need to find the angle s where sin(s) = √2/2.
From the unit circle, we know that sin(π/4) = √2/2. So one value of s in the given interval is s = π/4.
Since sin is positive in the given interval [π/2, π], there is another angle with the same sin value as π - π/4.
So the other value of s in the given interval is s = π - π/4 = 3π/4.
Therefore, the exact values of s in the interval [π/2, π] that have the circular function value sin(s) = √2/2 are s = π/4 and s = 3π/4.
2. [π/2, π]; cos(s) = -√3/2
On the unit circle, the point at angle s has coordinates (cos(s), sin(s)). Since cos(s) = -√3/2, we need to find the angle s where cos(s) = -√3/2.
From the unit circle, we know that cos(π/6) = √3/2. So one value of s in the given interval is s = π/6.
Since cos is negative in the given interval [π/2, π], there is another angle with the same cos value as -π/6.
So the other value of s in the given interval is s = π - π/6 = 5π/6.
Therefore, the exact values of s in the interval [π/2, π] that have the circular function value cos(s) = -√3/2 are s = π/6 and s = 5π/6.
To find the exact values of s in the given intervals that have the given circular function values, you will need to use the unit circle and the trigonometric identities.
1. [π/2, π]; sin s = √2/2
First, recall that the value of sin(pi/4) = √2/2. Since sin values are symmetric about the y-axis, the positive value of π/4 lies in the second quadrant, which is within the given interval [π/2, π]. So, one solution is s = π/4.
Next, note that sin(x) = 1/2 at π/6. However, this value is not within the given interval [π/2, π]. Therefore, there are no other solutions in this interval for sin s = √2/2.
2. [π/2, π]; cos s = -√3/2
Similar to the previous case, recall that cos(pi/6) = √3/2. Since cos values are symmetric about the y-axis, the positive value of π/6 lies in the first quadrant, which is not within the given interval [π/2, π]. Therefore, we need to find the equivalent angle in the second quadrant.
In the second quadrant, the adjacent side is negative. Therefore, the equivalent angle with a cosine value of -√3/2 is π - π/6 = 5π/6. This angle lies within the given interval. So, another solution is s = 5π/6.
To summarize:
1. For sin s = √2/2, within [π/2, π], s = π/4.
2. For cos s = -√3/2, within [π/2, π], s = 5π/6.