Download Examples 24 to 33

TRIGONOMETRIC RATIOS OF ANY ANGLE
Example 24:
Find the EXACT numeric value of sin 150o without using the calculator.
150o has a reference angle of 30o and we have memorized that sin 30o =
Therefore, sin 150o =
.
exactly.
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Question: Why is the numeric value negative?
Answer: Since 150o is a second-quadrant angle, 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 25:
Find the EXACT numeric value of tan 210o without using the calculator.
210o has a reference angle of 30o and we have memorized that tan 30o =
Therefore, tan 210o =
.
exactly.
***************************
Question: Why is the numeric value positive?
Answer: Since 210o is a third-quadrant angle, we know the numeric value must
be positive 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 26:
Find the EXACT numeric value of sin( 135o) without using the calculator.
135o has the same reference angle as 135o, that is, 45o and we have
memorized that sin 45o =
Therefore, sin ( 135o) =
***************************
.
exactly.
Question: Why is the numeric value negative?
Answer: Since 135o 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.
Reminder: Negative angles start along the positive x-axis in a coordinate
system, and then are drawn in clockwise direction. On the other hand,
positive angles start along the positive x-axis but are then drawn in counterclockwise direction.
Example 27:
Find the EXACT numeric value of cot 330o without using the calculator.
330o has a reference angle of 30o and we have memorized that cot 30o =
Therefore, cot 330o =
.
exactly.
***************************
Question: Why is the numeric value negative?
Answer: Since 330o 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 28:
Find the EXACT numeric value of sec 225o without using the calculator.
225o has a reference angle of 45o and we have memorized that sec 45o =
Therefore, sec 225o
.
exactly.
***************************
Question: Why is the numeric value negative?
Answer: Since 225o 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 29:
Find the EXACT numeric value of csc 660o without using the calculator.
660o = 360o + 300o . We can ignore 360o because it is equivalent to a complete
rotation about the Origin. On the other hand, 300o has a reference angle of 60o
and we have memorized that csc 60o =
Therefore, csc 660o =
.
exactly.
***************************
Question: Why is the numeric value negative?
Answer: Since 660o is a fourth-quadrant angle, ignoring the full rotation of 360o
and just using 300o, 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 30:
Find the EXACT numeric value of cos ( 570o) without using the calculator.
570o =
360o + ( 210o). We can ignore
360o because it is equivalent to a
complete rotation about the Origin. On the other hand, 210o has the same
reference angle as 210o , that is 30o and we have memorized that cos 30o =
.
Therefore, cos (-570o) =
exactly.
***************************
Question: Why is the numeric value negative?
Answer: Since 570o is a second-quadrant angle, ignoring the full rotation of
360o and just using 210o, 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.
Reminder: Negative angles start along the positive x-axis in a coordinate system, and then
are drawn in clockwise direction. On the other hand, positive angles start along the
positive x-axis but are then drawn in counter-clockwise direction.
Example 31:
Find the EXACT numeric value of cos 450o without using the calculator.
450o = 360o + 90o. We can ignore 360o because it is equivalent to a complete rotation
about the Origin. On the other hand, 90o is a Quadrantal Angle and those do not have
reference angles. Therefore, you must remember the following pattern, which is actually
copied from the chart in this lecture.
As you can see, cos 90o = 0. Therefore, cos 450o = 0 also.
Example 32:
Find the EXACT numeric value of sin ( 450o) without using the calculator.
450o =
360o + (-90o) . We can ignore
360o because it is equivalent to a complete
rotation about the Origin. On the other hand, ( 90o) is a Quadrantal Angle and those do
not have reference angles. Therefore, you must remember the following pattern, which is
actually copied from the chart in this lecture.
As you can see, sin ( 90o) =
1. Therefore, sin ( 450o) =
1 also.
Reminder: Negative angles start along the positive x-axis in a coordinate system, and then
are drawn in clockwise direction. On the other hand, positive angles start along the
positive x-axis but are then drawn in counter-clockwise direction.
Example 33:
Find the EXACT numeric value of tan 810o without using the calculator.
810o = 360o + 360o + 90o . We can ignore the two 360o angles because they are
equivalent to two complete rotations about the Origin. On the other hand, 90o is a
Quadrantal Angle and those do not have reference angles. Therefore, you must
remember the following pattern, which is actually copied from the chart in this lecture.
As you can see, tan 90o is undefined. Therefore, tan 810o is undefined
also.