Solve sin θ = 0.95 for -90º ≤ θ ≤ 90º.
if theta is between 0 and -90 (quadrant 4) the sin is negative. Therefore we only look at quadrant one.
sin^-1 0.95 = 71.8 degrees
To solve the equation sin θ = 0.95 for -90º ≤ θ ≤ 90º, we need to find the values of θ that satisfy the equation within the given range. Here's how we can do it:
Step 1: Recognize the special right triangle ratio
The sine function relates the ratio of the side opposite the angle to the hypotenuse of a right triangle. One special right triangle ratio is 0.95, which corresponds to the side opposite to an angle of approximately 71.78º in a right triangle with a hypotenuse of 1.
Step 2: Find the principal solution using inverse sine function
Using the inverse sine function (sin⁻¹), also known as arcsin, we can find the principal solution for sin θ = 0.95. Calculate sin⁻¹(0.95) using a scientific calculator, and it will give you the value of approximately 71.78º.
Step 3: Determine additional solutions within the given range
Since the given range is -90º ≤ θ ≤ 90º, we need to check if there are any other angles within this range that satisfy sin θ = 0.95.
The sine function is positive in the first and second quadrants, so we need to investigate the quadrants where θ lies. The principal solution lies in the first quadrant (0º < θ < 90º). To find the corresponding angle in the second quadrant (-90º < θ < 0º), we subtract the principal solution from 180º. Therefore, the corresponding angle in the second quadrant is approximately (180º - 71.78º) which is equal to 108.22º.
So, sin θ = 0.95 has two solutions within the given range of -90º ≤ θ ≤ 90º, which are approximately 71.78º and -108.22º.