write each of the following as trigonometric ratios of acute angles.

(a)sin160 degrees
(b)cos 310 degrees
(c)tan 200 degrees

I do these in 2 steps

1. in which quadrant is the angle, this determines the + or 1
2. How far is the angle from the x-axis, this gives me the "angle in standard position" , always an acute angle

so for b)
1. 310° is in IV, and in IV the cosine is positive. So cos 310 is positive.
2. 310 is 50° to the x-axis

so sin 310° = sin 50°

do the others the same way, and check results with a calculator.

(a) Well, sin(160 degrees) can be written as sin(180 degrees - 20 degrees). Since the sine function is positive in the second quadrant, and we're dealing with an acute angle, we can write this as sin(20 degrees).

(b) Similarly, cos(310 degrees) can be written as cos(360 degrees - 50 degrees). Since the cosine function is negative in the fourth quadrant, and we're dealing with an acute angle, we can write this as -cos(50 degrees).

(c) Finally, for tan(200 degrees), we can write it as tan(180 degrees + 20 degrees). Since the tangent function is positive in the third quadrant, and we're dealing with an acute angle, we can write this as tan(20 degrees).

(a) To express sin160 degrees as a trigonometric ratio of an acute angle, we need to find a reference angle in the first quadrant that is equivalent to 160 degrees. Since the sine function is positive in the second quadrant, we can subtract 180 degrees from 160 degrees to get the reference angle.

Reference angle = 160 degrees - 180 degrees = -20 degrees (negative because it is measured clockwise from the positive x-axis)

The reference angle, -20 degrees, is equivalent to 340 degrees in the standard position, which is in the first quadrant.

sin(-20 degrees) = sin(340 degrees) = sin(20 degrees)

So, sin160 degrees can be expressed as sin(20 degrees).

(b) To express cos310 degrees as a trigonometric ratio of an acute angle, we need to find a reference angle in the first quadrant that is equivalent to 310 degrees. Since the cosine function is positive in the fourth quadrant, subtracting 360 degrees from 310 degrees gives us the reference angle.

Reference angle = 310 degrees - 360 degrees = -50 degrees (negative because it is measured clockwise from the positive x-axis)

The reference angle, -50 degrees, is equivalent to 310 degrees in the standard position, which is in the fourth quadrant.

cos(-50 degrees) = cos(310 degrees)

So, cos310 degrees can be expressed as cos(-50 degrees).

(c) To express tan200 degrees as a trigonometric ratio of an acute angle, we need to find a reference angle in the first quadrant that is equivalent to 200 degrees. Since the tangent function is negative in the third quadrant, subtracting 180 degrees from 200 degrees gives us the reference angle.

Reference angle = 200 degrees - 180 degrees = 20 degrees

The reference angle, 20 degrees, is in the first quadrant.

tan(200 degrees) = tan(20 degrees)

So, tan200 degrees can be expressed as tan(20 degrees).

To write the trigonometric ratios of acute angles, we need to find the reference angle that is greater than 0 degrees but less than or equal to 90 degrees. Then we can determine the sign of the ratio based on the quadrant in which the angle lies.

(a) sin 160 degrees:
The reference angle for 160 degrees can be found by subtracting 180 degrees: 160 degrees - 180 degrees = -20 degrees.
Since the reference angle is negative, and sine is positive in the second quadrant, the trigonometric ratio is positive.
So, sin 160 degrees is equivalent to sin(-20 degrees).

(b) cos 310 degrees:
The reference angle for 310 degrees can be found by subtracting 360 degrees: 310 degrees - 360 degrees = -50 degrees.
Since the reference angle is negative, and cosine is positive in the fourth quadrant, the trigonometric ratio is positive.
So, cos 310 degrees is equivalent to cos(-50 degrees).

(c) tan 200 degrees:
The reference angle for 200 degrees can be found by subtracting 180 degrees: 200 degrees - 180 degrees = 20 degrees.
Since the reference angle is positive, and tangent is negative in the third quadrant, the trigonometric ratio is negative.
So, tan 200 degrees is equivalent to -tan(20 degrees).

Remember that the sine, cosine, and tangent values of an angle do not change if the angle is in standard position or if it is a negative angle. The sign of the trigonometric ratio depends on the quadrant in which the angle lies.