Determine the exact value of the following.
a) sin300 degress
b) tan (-135 degrees)
c)csc150 degrees
d) sec45 degrees
e) cos 4pi/3
f) sin (-3pi/4)
draw the triangles in standard position. They are all either 45-45-90 of 30-60-90.
Those values you know, right?
To determine the exact value of trigonometric functions, we can use the unit circle and the reference angles. The unit circle is a circle with a radius of 1 unit centered at the origin (0, 0) in the coordinate plane.
a) To find the value of sin(300 degrees):
Step 1: Convert the angle to a reference angle.
Since the unit circle covers 360 degrees or 2π radians, we need to subtract 360 degrees from 300 degrees to find the reference angle: 300 - 360 = -60 degrees.
Step 2: Determine the quadrant of the angle.
Since -60 degrees falls in the fourth quadrant, the sine value will be negative.
Step 3: Find the sine value of the reference angle.
The reference angle for -60 degrees is 60 degrees. The sine of 60 degrees is √3/2.
Step 4: Apply the sign based on the quadrant.
Since -60 degrees falls in the fourth quadrant, the final result will be -√3/2.
Therefore, sin(300 degrees) = -√3/2.
b) To find the value of tan(-135 degrees):
Step 1: Convert the angle to a reference angle.
Since the unit circle covers 360 degrees or 2π radians, we need to add 360 degrees to -135 degrees to find the reference angle: -135 + 360 = 225 degrees.
Step 2: Determine the quadrant of the angle.
Since 225 degrees falls in the third quadrant, the tangent value will be positive.
Step 3: Find the tangent value of the reference angle.
The reference angle for 225 degrees is 45 degrees. The tangent of 45 degrees is 1.
Step 4: Apply the sign based on the quadrant.
Since 225 degrees falls in the third quadrant, the final result will be 1.
Therefore, tan(-135 degrees) = 1.
c) To find the value of csc(150 degrees):
Step 1: Convert the angle to a reference angle.
Since the unit circle covers 360 degrees or 2π radians, we need to subtract 360 degrees from 150 degrees to find the reference angle: 150 - 360 = -210 degrees.
Step 2: Determine the quadrant of the angle.
Since -210 degrees falls in the third quadrant, the cosecant value will be negative.
Step 3: Find the cosecant value of the reference angle.
The reference angle for -210 degrees is 30 degrees. The cosecant of 30 degrees is 2.
Step 4: Apply the sign based on the quadrant.
Since -210 degrees falls in the third quadrant, the final result will be -2.
Therefore, csc(150 degrees) = -2.
d) To find the value of sec(45 degrees):
Step 1: Determine the quadrant of the angle.
Since 45 degrees falls in the first quadrant, the secant value will be positive.
Step 2: Find the secant value of the angle.
The secant of 45 degrees is √2.
Therefore, sec(45 degrees) = √2.
e) To find the value of cos(4π/3):
Step 1: Convert the angle to degrees.
To convert radians to degrees, use the formula: degrees = radians * (180/π)
4π/3 * (180/π) = 240 degrees.
Step 2: Determine the quadrant of the angle.
Since 240 degrees falls in the third quadrant, the cosine value will be negative.
Step 3: Find the cosine value of the angle.
The cosine of 240 degrees is -1/2.
Therefore, cos(4π/3) = -1/2.
f) To find the value of sin(-3π/4):
Step 1: Convert the angle to degrees.
To convert radians to degrees, use the formula: degrees = radians * (180/π)
-3π/4 * (180/π) = -135 degrees.
Step 2: Determine the quadrant of the angle.
Since -135 degrees falls in the second quadrant, the sine value will be positive.
Step 3: Find the sine value of the angle.
The sine of 135 degrees is √2/2.
Therefore, sin(-3π/4) = √2/2.