express the followinf trig functions in terms of an acute angle.
(a) cos 143 degrees
(b) tan 192 degrees
I do these with the following steps:
cos 143
- which quadrant is it in? --- 2nd
- by CAST rule, what sign is the cosine in the 2nd? --- negative
- how far from 143º to the x-axis ? --- 37
so cos 143 = - cos 37
do the other one the same way.
To express the trigonometric functions in terms of an acute angle, we will use the trigonometric identities for reference angles.
For part (a):
(a) cos 143 degrees
To find the reference angle, subtract 180 degrees from 143 degrees:
Reference angle = 143 degrees - 180 degrees = -37 degrees
Since the cosine function is positive in the fourth quadrant, we can say that cos 143 degrees is equal to cos(-37 degrees).
For part (b):
(b) tan 192 degrees
To find the reference angle, subtract 180 degrees from 192 degrees:
Reference angle = 192 degrees - 180 degrees = 12 degrees
To express tan 192 degrees in terms of an acute angle, we can say that tan 192 degrees is equal to tan 12 degrees.
Therefore, the expressions in terms of an acute angle are:
(a) cos 143 degrees = cos(-37 degrees)
(b) tan 192 degrees = tan 12 degrees
To express trigonometric functions in terms of an acute angle, we can use the unit circle and the periodicity of the trigonometric functions. Here's how you can do it for the given trig functions:
(a) cos 143 degrees:
1. To start, we need to find an acute angle that is equivalent to 143 degrees. Since the cosine function has a period of 360 degrees, we can subtract multiples of 360 until we obtain an acute angle.
143 degrees - 360 degrees = -217 degrees (not acute)
-217 degrees + 360 degrees = 143 degrees (acute)
So, cos 143 degrees is equivalent to cos 143 degrees = cos(143 degrees - 360 degrees) = cos(-217 degrees).
2. Next, we need to find the reference angle for -217 degrees. The reference angle is the positive acute angle formed between the terminal side of the angle and the x-axis. Since the unit circle is symmetric across the x-axis, the reference angle for -217 degrees is the same as the reference angle for 217 degrees.
So, the reference angle for 217 degrees is 217 degrees.
3. Now, we need to determine the sign of the cosine function. The cosine function is positive in the first and fourth quadrants, and negative in the second and third quadrants. Since -217 degrees is in the third quadrant, the cosine function will be negative.
4. Finally, we can express cos 143 degrees in terms of an acute angle as follows:
cos 143 degrees = -cos 217 degrees
(b) tan 192 degrees:
1. Similarly, we begin by finding an acute angle equivalent to 192 degrees. Since the tangent function has a periodicity of 180 degrees, we can subtract multiples of 180 until we obtain an acute angle.
192 degrees - 180 degrees = 12 degrees (acute)
So, tan 192 degrees is equivalent to tan 192 degrees = tan(192 degrees - 180 degrees) = tan 12 degrees.
2. We don't need to find a reference angle for tangent since it is defined as the ratio of sine and cosine, rather than a distance on the unit circle.
So, tan 192 degrees can be expressed in terms of an acute angle as tan 12 degrees.