How would you go about solving something like sin(5pi/3) without a calculator? Would you use a unit circle?
yes
or knowing that 5π/3 radians is 300°
the angle in standard position is 60°, that is, if I construct the triangle with
60° in the first quadrant.
You should know from your unit circle that sin60° = √3/2
But 300° is in quadrant III, and in that quad the sine is negative, thus....
sin 5π/3 = -√3/2
Btw, knowing the ratios of sides in a standard 30-60-90 and a 45-45-90 degree triangle means you don't even need the unit circle.
To solve trigonometric functions without a calculator, using a unit circle is indeed a good approach. Here's how you can solve sin(5π/3) step by step:
1. Start by understanding the unit circle: A unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is used to find the values of trigonometric functions for different angles.
2. Identify the given angle: In this case, the given angle is 5π/3. Angles in the unit circle are usually measured counterclockwise from the positive x-axis.
3. Determine the reference angle: The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. To find the reference angle for 5π/3, subtract it from 2π (a full revolution): 2π - (5π/3) = (6π/3) - (5π/3) = (1π/3). Therefore, the reference angle is π/3.
4. Determine the quadrant: The quadrant can be determined by the signs of the x and y coordinates in the unit circle. Since sin is positive in both the first and second quadrants, we need to determine whether sin(π/3) is positive or negative.
5. Evaluate the sine of the reference angle: Look for the point on the unit circle corresponding to the reference angle of π/3. In the first and second quadrants, the x-coordinate of this point is always positive. Therefore, sin(π/3) is positive.
6. Determine the value of sin(5π/3): Since sin(5π/3) is in the same quadrant as sin(π/3) and has the same value, the answer is positive. So, sin(5π/3) = sin(π/3) = √3/2.
By following these steps and using the unit circle, you can calculate trigonometric functions without the need for a calculator.