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Illustrative
Mathematics
F-TF Trigonometric functions
for arbitrary angles (radians)
Alignments to Content Standards: F-TF.A.2
Task
Below is a picture of a right triangle with
Joyce knows that the sine of
the hypotenuse :
a the measure of angle A:
a is the length of the side opposite A divided by length of
sin a =
|BC|
.
|AC|
Joyce says, ''the sine of an obtuse angle does not make any sense because I can't make
a right triangle with an obtuse angle.''
a. Draw a picture and explain how Joyce might define the sine of an obtuse angle.
b. What are
sin
3π
and
4
sin π? Why?
IM Commentary
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Illustrative
Mathematics
The purpose of this task is to examine trigonometric functions for obtuse angles. The
values sin x and cos x are defined for acute angles by referring to a right triangle one of
whose acute angles measures x. For an obtuse angle, no such triangle exists and so an
alternate definition is required.
Prior to working on this task, students should have experience working with
trigonometric functions and how they relate to the unit circle. The task provides a
means to assess student learning of these definitions. (Another task using degrees, F-TF
Trigonometric functions for arbitrary angles (degrees), can be used to introduce the
idea.) One advantage to working with points on the unit circle is that sine and cosine of
angles like 0 and π still make sense even though they do not belong to any triangle.
Edit this solution
Solution
a. Joyce is right that there is no right triangle with an angle of measure a if a > 90. She
is also right that that the definition of sin a in terms of right triangles does not make
sense. Instead, we can begin by drawing an angle of measure a, inscribed in a circle of
−→
radius 1, with ray AB along the positive x-axis:
Consider an acute angle BAD of measure θ, inscribed in a circle,
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Illustrative
Mathematics
The coordinates of D are (cos θ, sin θ). We can use this fact to make sense of sin a and
cos a for the obtuse angle a. In other words, for the obtuse angle a we define sin a and
cos a so that
C = (cos a, sin a).
Looking at the picture, we can see that for angles with measure 90
have sin a > 0 and cos a < 0.
< a < 180, we will
b. The circumference of the unit circle is 2π so an angle of π/2 is a right angle and π/4 is
half of a right angle or 45 degrees. Below is a picture of an angle of measure 3π/4:
If the purple circle is a unit circle, then from part (a) we have sin 3π/4 = |CD| and
cos 3π/4 = −|AD|. We have that △ADC is a 45-45-90 triangle and so
C = (−√2‾/2, √2‾/2). So we have sin 3π/4 = √2‾/2 and cos 3π/4 = −√2‾/2.
The circumference of the unit circle is 2π so an angle of π is half of the circle or 180
degrees. An angle of measure π is picured below and it is a line:
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Illustrative
Mathematics
The coordinates of C are (-1,0) so sin π
= 0 and cos π = −1.
F-TF Trigonometric functions for arbitrary angles (radians)
Typeset May 4, 2016 at 21:17:23. Licensed by Illustrative Mathematics under a
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