Given θ = arcsin (tan45°) find the exact degree measure of θ without usiing calculator.
tan 45° = 1
θ = arcsin(1) = 90°
Why did the angle go to therapy? Because it was feeling a little "sinful" and wanted to find its true identity. Okay, let's digress from the jokes. Let's solve this problem!
To find the exact degree measure of θ without using a calculator, we'll use some trigonometric identities and properties.
First, we'll recognize that tan45° = 1. This is because tan (45°) is the ratio of the length of the side opposite to the angle (which is 1) to the length of the adjacent side (also 1).
Next, we'll use the identity that sin(θ) = tan(θ) / sqrt(1 + tan²(θ)).
Since we know that tan45° = 1, we can substitute it into the identity:
sin(θ) = 1 / sqrt(1 + 1²) = 1 / sqrt(2).
Lastly, we'll find the angle whose sine is equal to 1 / sqrt(2). This angle is 45°, which means θ = 45°.
So, the exact degree measure of θ is 45°. And hey, that's pretty easy to remember, right?
To find the exact degree measure of θ, we need to simplify the expression arcsin(tan45°) without using a calculator.
First, let us consider the value of tan(45°). Since tan(45°) = 1, we can substitute it into the expression:
θ = arcsin(1)
The arcsin function gives us the angle whose sine is equal to the given value. In this case, we are looking for an angle whose sine is equal to 1.
The angle whose sine is equal to 1 is 90°. Therefore, the exact degree measure of θ is 90°.
To solve this problem without a calculator, we can use the fact that the cosine function is the reciprocal of the secant function.
First, let's find the value of tan(45°):
tan(45°) = 1
Using the definitions of sine and cosine:
sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
We know that sin(θ) = tan(θ)/sec(θ).
Therefore, we can rewrite the equation as:
sin(θ) = tan(45°)/sec(θ)
We need to find the angle whose sine is equal to the value we obtained. Since the sine function is positive in both the first and second quadrants, we need to consider both possibilities.
First, consider the first quadrant (0° to 90°):
In the first quadrant, sin(θ) = tan(45°)/sec(θ) = 1/sec(θ).
To find the secant of an angle θ, we can use the Pythagorean identity which states that sec(θ) = 1/cos(θ).
Therefore, sin(θ) = 1/(1/cos(θ)) = cos(θ).
This means that sin(θ) = cos(θ) = 1.
So, in the first quadrant, the angle whose sine is equal to 1 is 90°.
Now let's consider the second quadrant (90° to 180°):
In the second quadrant, the sine function is positive, but the cosine function is negative.
Since sin(θ) = cos(θ) = 1 and the cosine function is negative in the second quadrant, the angle θ in the second quadrant must be 180° - 90° = 90°.
So, the exact degree measure of θ is 90°.