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Illustrative
Mathematics
G-SRT Tangent of Acute Angles
Alignments to Content Standards: G-SRT.C.6
Task
a. Below is a picture of a right triangle:
In terms of the picture, what are sin ∠P, cos ∠P, and
on the size of the triangle?
tan ∠P? Do these values depend
b. Complete the following table, rounding off each answer to the nearest hundredth if
using a calculator. Draw a picture showing the meaning of sin x, cos x, and tan x for an
acute angle x.
Angle (degrees)
cos x
sin x
tan x
0
1
Illustrative
Mathematics
15
30
45
60
75
90
c. What value do you find in the bottom right corner of the table for
tan 90? Why?
d. What patterns do you notice in the third column with the values of
think that these patterns will hold true for all acute angles x? Explain.
e. Based on the table, what values do you think the function
0 < x < 90? Explain.
tan x? Do you
tan x takes when
IM Commentary
The purpose of this task is to focus on studying values of tan x for special angles and
conjecturing from these values how the function tan x varies when 0 ≤ x < 90. This
task complements https://www.illustrativemathematics.org/illustrations/1868 which
studies the same table for sin x and cos x but looks at some other expressions in terms
of sin x and cos x rather than tan x. The teacher might also wish to add another column
for the cotangent function as students should be able to conjecture and show that
tan x = cot (90 − x) for acute angles x. Exact values for sin x and cos x can be found in
the commentary to the aforementioned task and these could be used to obtain exact
values of tan x and cot x. Our practice in the solution to this task is to put in exact
values for benchmark angles and approximate decimal values for the non benchmark
angles. Exact such values can be found, for example, from half-angle formulas applied
to known values (e.g., applying the sine half-angle formula to compute sin(15∘ ) from
sin(30∘ )).
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Illustrative
Mathematics
Edit this solution
Solution
a. The sine of
|PR|:
∠P is the length of the side opposite P, |QR|, divided by the hypotenuse,
sin P =
|QR|
.
|PR|
The cosine of ∠P is the length of the side adjacent to
the hypotenuse, |PR|:
cos P =
|PQ|
.
|PR|
The tangent of ∠P is the length of the side opposite
adjacent to P, |PQ|:
tan P =
P, |PQ|, divided by the length of
P, |QR| divided by the side
|QR|
.
|PQ|
These ratios do not depend on the size of the triangle. If the triangle is scaled by a
(positive) factor of r then all three side lengths scale by r. The three trigonometric
ratios are scaled by a factor of rr = 1 and so they do not depend on the size of the
triangle.
b. Exact values are entered in each row except for 15∘ and 75∘ :
Angle (degrees)
cos x
sin x
tan x
0
1
0
0
15
0.97
0.26
0.27
30
√3
2
0.5
1
√3
45
√2
2
√2
2
1
60
0.5
√3
2
√3‾
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Illustrative
Mathematics
75
0.26
0.97
3.7
90
0
1
--
Note that there is no entry for tan 90 because 1 is not defined.
0
c. We have tan x
=
sin x
and this function is not defined when the denominator,
cos x
is zero. This happens, on the unit circle, for x
0 ≤ x < 360, tan x has a well defined value.
cos x,
= 90 and x = 270. For all other angles
d. There are a few patterns. First, the values of tan x for 0 ≤ x < 90 appear to be nonnegative and the value increases as x increases. Further experimentation will show that
this pattern continues. For example, tan 80 ≈ 5.7, tan 85 ≈ 11, and tan 89 ≈ 57. This
makes sense since in a right triangle with an 89 degree angle and a 1 degree angle, the
side opposite the 89 degree angle is much larger than the side adjacent and this
disparity grows as the 89 angle grows closer and closer to 90 degrees.
Another pattern is perhaps harder to see. We can see that
had exact values, we also have tan 75
=
tan 60 =
1
and, if we
tan 30
1
. In general for a (non-zero) acute angle
tan 15
1
x, we have tan x = tan (90−x)
. This comes from the identities sin x = cos (90 − x) and
cos x = sin (90 − x). To see why, note that
sin x
cos x
cos (90 − x)
=
sin (90 − x)
1
=
tan (90 − x)
tan x =
e. We can see that tan 0 = 0 and we have also seen that for 0 ≤ x < 90, tan x grows as
x increases. There is no bound to how big tan x can be because when we write it as
sin x
we can see that for acute angles just a little bit less than 90 degrees, the
cos x
numerator of this fraction, sin x, is close to 1 while the denominator, cos x is very close
to zero. So the range of tan x for 0 < x < 90 is all non-negative real numbers.
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Illustrative
Mathematics
G-SRT Tangent of Acute Angles
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