Calculate the exact values of sin(theta) cos(theta) and tan (theta) when theta=240 degrees.
How would I do this with and without a calculator?
Learn your standard angles in their various orientation around the circle.
0 pi/6 pi/4 pi/3 and pi/2 are important enough to memorize
In degrees, that's 0 30 45 60 90
240 = 180 + 60
so it's in the 3rd quadrant.
Use the functions of 60 degrees, except that sin and cos are both negative there.
To calculate the exact values of sin(theta), cos(theta), and tan(theta) when theta = 240 degrees, we can use the concept of reference angles and basic trigonometric identities.
Using Reference Angles:
1. Convert theta from degrees to radians: theta_radians = (theta * pi) / 180 = (240 * pi) / 180 = 4pi/3 radians.
2. Find the reference angle, which is the acute angle formed between the terminal side of the angle and the x-axis in standard position. In this case, the reference angle is 60 degrees (240 - 180).
3. Identify the quadrant in which the angle lies. In this case, 240 degrees is in the third quadrant.
4. Now, based on the quadrant, assign the appropriate signs to sine, cosine, and tangent:
- sine (sin): In the third quadrant, sin(theta) value is negative.
- cosine (cos): In the third quadrant, cos(theta) value is negative.
- tangent (tan): In the third quadrant, tan(theta) value is positive.
5. Use the reference angle to determine the exact values of sin(theta), cos(theta), and tan(theta) using the following trigonometric identities:
- sin(theta) = -sin(reference angle) = -sin(60 degrees) = -sqrt(3)/2
- cos(theta) = -cos(reference angle) = -cos(60 degrees) = -1/2
- tan(theta) = tan(reference angle) = tan(60 degrees) = sqrt(3)
Without a Calculator:
To calculate the values without using a calculator, you need to memorize or have access to trigonometric tables that provide the exact values of sine, cosine, and tangent for commonly used angles.
Using a Calculator:
1. Enter 240 on your calculator.
2. Calculate sin(240), then cos(240), and finally tan(240). Make sure your calculator is in degree mode.
3. The calculator will provide the approximate values of sin(theta), cos(theta), and tan(theta) when theta=240 degrees.
Note: While using a calculator yields approximate values, to find the exact values, you must use the reference angle approach explained earlier.