Download 1 - Manhasset Public Schools

Aim #27: What is the relationship between the sine and cosine
of complementary angles and special angles ?
1-21-15
Do Now: Given the diagram of the right triangle, complete the following table.
Calculate the hypotenuse in simplest radical form. Express all side lengths in
simplest radical form with rational denominators.
Angle Measure sin θ
cos θ
tan θ
s
2
s
t
4
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).
α and β are a pair of _____________________ angles
B
Determine the following values in the table.
c
A
a
b
C
Based on the results in the above table, what can you conclude ?
• 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).
Any two complementary angles can be the two ________ angles in a right triangle.
The co- prefix in cosine refers to the fact that the cosine of an angle
equals the sine of its ___________________.
Example 1:
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 BC two different ways.
AB
c) Use trig. ratios to express AC two different ways.
AB
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)
3. For what angle measurement must sine and cosine have the same value.
Explain how you know.
Example 2: 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____________________________________.**
Example 3: What are the exact sine and cosine values for the "special angles"?
a) Write in the sine and cosine value for 00 and 900 determined in Example (2).
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
Example 4: The following diagrams show the basic side lengths of the 30-60-90
and 45-45-90 triangles.
600
450
2
1
1
300
450
1
Determine side lengths of three different triangles similar to each of the
triangle in the chart. Then generalize the side lengths in terms of x.
30­60­90 Triangle
side lengths 1: : 2
45­45­90 Triangle
side lengths 1: 1 : ______ : ______ : ______
______ : ______ : ______
______ : ______ : ______
______ : ______ : ______
______ : ______ : ______
______ : ______ : ______
x : ______ : ______
x : ______ : ______
Exercises
1. The triangles below are 30-60-90 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
450
c
450
a
c
a
450
2. Given an equilateral triangle with sides of length 9, find the length of the
altitude.
Confirm your answer using the Pythagorean Theorem.
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 #27
#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 30-60-90 right triangles. Find the unknown
lengths x and y, using sin and cos values of an acute angle. Show your solving of an
appropriate equation.
9.
8.
7.
y
300
600
600
y
7
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 7√2. Use sine or cosine to find the length of the
diagonal of the square. Confirm your answer using the Pythagorean Theorem.
7√2
7√2
14a) Make a prediction about how the sum sin 30 + cos 60 will relate to the
sum sin 60 + cos 30.
b) Use the sine and cosine values of special angles to find the sum: sin 30 + cos 60.
c) Find the exact value of the sum: sin 60 + cos 30.
d) Was your prediction correct ? Explain why or why not.