Download TRIGONOMETRIC RATIOS OF ANY ANGLE Example 34: Find the

TRIGONOMETRIC RATIOS OF ANY ANGLE
Example 34:
Find the EXACT numeric value of cos (5 /3) without using the calculator.
5 /3 has a reference angle of
Therefore, cos (5 /3) =
/3 and we have memorized that cos
/3 =
.
exactly.
Question: Why is the numeric value positive?
Answer: Since 5 /3 is a fourth-quadrant angle, we know the numeric
value must be positive after using All Students Take Calculus. That is,
in the fourth quadrant all trigonometric ratios have negative numeric
values except for cosine and secant.
Example 35:
Find the EXACT numeric value of tan (5 /6) without using the calculator.
5 /6 has a reference angle of
Therefore, tan (5 /6) =
/6 and we have memorized that tan
/6 =
exactly.
Question: Why is the numeric value negative?
Answer: Since 5 /6 is a second-quadrant angle, we know the numeric
value must be negative after using All Students Take Calculus. That is,
in the second quadrant all trigonometric ratios have negative numeric
values except for sine and cosecant.
Example 36:
Find the EXACT numeric value of cos ( 3 /4) without using the calculator.
3 /4 has the same reference angle as 3 /4, that is,
memorized that cos ( /4) =
.
Therefore, cos ( 3 /4) =
exactly.
/4 and we have
.
Question: Why is the numeric value negative?
Answer: Since 3 /4 is a third-quadrant angle, we know the numeric
value must be negative after using All Students Take Calculus. That is,
in the third quadrant all trigonometric ratios have negative numeric values
except for tangent and cotangent.
Example 37:
Find the EXACT numeric value of cot (11 /6) without using the calculator.
11 /6 has a reference angle of
/6 and we have memorized that cot
/6 =
.
Therefore, cot (11 /6) =
exactly.
Question: Why is the numeric value negative?
Answer: Since 11 /6 is a fourth-quadrant angle, we know the numeric
value must be negative after using All Students Take Calculus. That is,
in the fourth quadrant all trigonometric ratios have negative numeric
values except for cosine and secant.
Example 38:
Find the EXACT numeric value of sec (7 /4) without using the calculator.
7 /4 has a reference angle of
Therefore, sec (7 /4) =
/4 and we have memorized that sec
/4 =
exactly.
Question: Why is the numeric value positive?
Answer: Since 7 /4 is a fourth-quadrant angle, we know the numeric
value must be positive after using All Students Take Calculus. That is,
in the fourth quadrant all trigonometric ratios have negative numeric
values except for cosine and secant.
Example 39:
Find the EXACT numeric value of sin (8 /3) without using the calculator.
Since 8 /3 = 6 /3 + 2 /3, we use the 2 /3 angle (which is coterminal with the 8 /3
angle) to find the reference angle.
Remember we just learned that coterminal angles have the same Reference Angle!
The 2 /3 angle is a second-quadrant angle with a reference angle of
This is also the reference angle of the 8 /3 angle.
2 /3 =
/3.
We have memorized that sin
Therefore, sin (8 /3) =
/3 =
.
.
Question: Why is the numeric value positive?
Answer: Since 8 /3 is a second-quadrant angle, ignoring the full rotation
of 6 /3 = 2 and just using 2 /3, we know the numeric value must be
positive after using All Students Take Calculus. That is, in the second
quadrant all trigonometric ratios have negative numeric values except for
sine and cosecant.
Example 40:
Find the EXACT numeric value of cos ( 17 /6) without using the calculator.
Since 17 /6 = 12 /6 + ( 7 /6), we use the 7 /6 angle (which is
coterminal with the 17 /6 angle) to find the reference angle.
Remember we just learned that coterminal angles have the same Reference
Angle .
Furthermore, the Reference Angle of a negative angle is the same as its
positive counterpart.
Therefore, we use the 7 /6 angle which is a third-quadrant angle with a
reference angle of 7 /6
= /6.
This is also the reference angle of the
We have memorized that cos ( /6) =
Therefore, cos ( 17 /6) =
17 /6 angle.
.
.
Question: Why is the numeric value negative?
Answer: Since 17 /6 is a second-quadrant angle, ignoring the full
rotation of 2 and just using 7 /6, we know the numeric value must
be negative after using All Students Take Calculus. That is, in the
second quadrant all trigonometric ratios have negative numeric values
except for sine and cosecant.
Example 41:
Find the EXACT numeric value of cos (3 ) without using the calculator.
3 = 2 + . We can ignore 2 because it is equivalent to a complete rotation about
the Origin. On the other hand, is a Quadrantal Angle which does not have a reference
angle. Therefore, you must remember the following pattern, which is actually copied
from the chart in this lecture.
As you can see, cos
=
1. Therefore, cos (3 ) =
1 also.
Example 42:
Find the EXACT numeric value of sin ( 5 ) without using the calculator.
5 =( 2 )+( 2 )+(
). We can ignore the two 2 angles because they are
equivalent to two complete rotations about the Origin. On the other hand,
is a
Quadrantal Angle which does not have a reference angle. Therefore, you must
remember the following pattern, which is actually copied from the chart in this lecture.
As you can see, sin (
) = 0. Therefore, sin ( 5 ) = 0 also.
Example 43:
Find the EXACT numeric value of tan (4 ) without using the calculator.
4 = 2 + 2 + 0. We can ignore the two 2 angles because they are equivalent to two
complete rotations about the Origin. On the other hand, 0 is a Quadrantal Angle which
does not have a reference angle. Therefore, you must remember the following pattern,
which is actually copied from the chart in this lecture.
As you can see, tan (0) = 0. Therefore, tan (4 ) = 0 also.