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CC Geometry H
Aim #24: What is the relationship between the sine and cosine
of complementary angles?
Do Now: Given the diagram of the right triangle, complete the following table.
Express all ratios in simplest radical form.
Angle Measure sin θ
cos θ
tan θ
s
2
s
4
t
t
a) Which values are equal?
b) How are tan t and tan s related?
How are the two co-functions (sine and cosine) related?
In right triangle ABC, the measurement of acute angle ≮A is denoted by α (alpha),
and the measurement of acute angle ≮B is denoted by β (beta).
B
α and β are a pair of _____________________ angles
Determine the following ratios:
c
A
a
b
C
• Since the ratios for sine α & cosine β are the same, ________ = ________.
• Since the ratios for cosine α & sine β are the same, ________ = ________ .
The sine of an angle is equal to the ____________ of its ________________,
and the______________ of an angle is equal to the ______ of its complement.
For complementary angles α and β, sin α = cos ____ and cos ___ = sin β.
Given measure θ such that 0 < θ < 90, cos (θ) = sin (90 - θ) and sin θ = cos (90 - θ).
The two complementary angles in a right triangle are the ________ angles.
The co- prefix in cosine refers to the fact that the cosine of an angle
equals the sine of its ___________________.
Consider right triangle ABC with right ≮C, and the degree measures of ≮A and ≮B
are α and β, respectively.
a) Find α + β. _____
b) Use trig. ratios to express
c) Use trig. ratios to express
two different ways.
5
4
two different ways.
3
d) What can you conclude about sin α and cos β ?
e) What can you conclude about cos α and sin β ?
2. Find value of θ that make each statement true.
a) sin θ = cos (25)
b) sin 80 = cos θ
c) sin θ = cos (θ + 10)
d) cos(θ - 45) = sin θ
e) cos θ = sin (θ - 30)
f) sin (θ + 20) = sin ( θ + 40)
3. For what angle measurement must sine and cosine have the same value. Explain
how you know.
What are sine and cosine values for θ = 00 and 900 ?
In the figure to the right, the hypotenuse c
of right ΔABC is the radius of a circle
which has a radius of length 1 unit.
1
• c = ___
• sin θ =
=
and cos θ =
=
.
As θ gets closer to 00, a __________ and sin θ approaches ___.
As θ gets closer to 00, b __________ and cos θ approaches ___.
Definitions: sin 00 = _____ and cos 00 = _____
As θ gets closer to 900, a __________ and gets closer to ___, and the value of
sin θ approaches ___.
As θ gets closer to 900, b __________ and gets closer to ___, and the value of
cos θ approaches ___.
Definitions: sin 900 = _____ and cos 900 = _____
**Since sin 00 = cos 900, and cos 00 = sin 900, this is consistent with the fact that
the sine of an angle equals the_____________________________________.**
What are the exact sine and cosine values for the "special angles"?
a) Write in the sine and cosine value for 00 and 900:
θ
00
300
600
450
900
sin θ
cos θ
b) Determine the exact sine and cosine value for 300, 600, and 450 using the
equilateral triangle with side 2 and isosceles right triangle with side 1 below.
Write your answers, with rationalized denominators, in the chart above.
450
300
2
1
600
1
**Memorize the values of sine and cosine for the special angles above.**
To help memorize, note that the complementary angles have the same values but
in reverse order.
•
sin 00 = cos 900 and sin 900 = cos 00
•
sin 30 = cos 600 and sin 600 = cos 300
•
sin 450 = cos 450
θ
00
450
300
900
600
sin θ
cos θ
1. The triangles below are special right triangles. Find the unknown lengths a and
b, using sin and cos values of an acute angle. Show your solving of equations.
a)
b)
3
a
c
3
300
b
600
a
c)
e)
d)
a
b
45
c
450
c
a
0
450
a
Let's Sum it Up!
• The sine of an angle is equal to the cosine of its complementary angle, and the
cosine of and angle is equal to the sine of its complementary angle.
• Sin 900 = 1 and cos 00 = 1 and similarly, sin 00 = 0 and cosine of 900 = 1.
• The values for the cosine and sine values for the special angles are the same,
but they are in reverse order.
Name_____________________
Date _____________________
CC Geometry H
HW #24
#1-6 Find the values of θ that make the equation true.
1. sin θ = cos 32
2. cos 11 = sin θ
3. sin θ = cos (θ + 38)
3. sin (θ + 10) = sin 60
4. cos θ = sin (3θ + 20)
6.
#7-12 The triangles below are special right triangles. Find the unknown lengths
using sin and cos values of an acute angle. Show your solving of an appropriate
equation.
7.
9.
8.
600
300
600
y
7
y
x
x
x
y
12
11.
10.
12.
10
x
450
y
450
x
x
45
0
y
y
OVER
13. A square has side lengths
. Use sine or cosine to find the length of the
diagonal of the square. Confirm your answer using the Pythagorean Theorem.
14. Use the sine and cosine values of special angles to find the exact sum of:
a) sin 30 + cos 60
b) sin 60 + cos 30
Mixed Review:
1) Given an equilateral triangle with sides of length 9, find the length of the
altitude.
2) Which statement describe the properties of a rhombus? Select all that
apply.
a. The diagonals bisect the angles.
b. The diagonals bisect each other.
c. The opposite sides are parallel.
d. The opposite angles are congruent.
e. The diagonals are congruent.
f. The diagonals are perpendicular.