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x ≈ 13.79
7.1√2
Aim #23: What are the sine, cosine and tangent ratios?
Do Now:
a) Write the ratio of opp/hyp for ≮A and ≮B.
CC Geometry H
A
5
b) Write the ratio of adj/hyp for ≮A and ≮B.
C
13
B
12
c) What do you notice about the relationship between these ratios for ≮s A and B?
Three Trigonometric Ratios
If
sine A =
A is an acute angle in a right triangle, then we say:
opposite
hypotenuse
cosine A =
opposite
adjacent
tangent A =
adjacent
hypotenuse
1) In right ΔPQR, m≮P = 53.2o and m≮Q = 36.8o. Complete the following table.
Measure of Angle
Sine (opp/hyp)
Cos (adj/hyp)
Tan (opp/adj)
53.2o
36.8o
2) In right ΔABC, m≮A = 33.7o and m≮B = 56.3o. Complete the following table.
Measure of Angle
Sine (opp/hyp)
Cos (adj/hyp)
Tan (opp/adj)
33.7o
56.3o
*To notice patterns, we will not rationalize our denominators in today's lesson.
3) Complete the following table.
(opp/hyp)
(adj/hyp)
(opp/adj)
4) Complete the following table.
(opp/hyp)
(adj/hyp)
(opp/adj)
5) Tammy did not finish completing the table below for a diagram similar to the
previous problems that the teacher had on the board where p was the measure of
angle P and q was the measure of angle Q. Use any patterns you notice to
complete the table.
6) If u and v are the measures of complementary angles such that sin u = 25 and
tan v =
, label the sides of the right triangle in the diagram below with possible
lengths.
We will be using the trig functions on our calculators to get ratios for the given
angles. Make sure you are in DEGREE MODE.
7) A fisherman is at pt F on the open sea and has three favorite fishing
locations. The locations are indicated by A, B and C. The fisherman plans to sail
from F to A, then to B, then to C, then back to F. If the fisherman is 14 miles
from AC, find the total distance that he will sail to the nearest tenth of a mile.
8) The triangles below contain approximations for all lengths. Find sin A, cos
A, and tan A. Compare these ratios to sin D, cos D, and tan D. What do you
notice?
These two right triangles which each have an acute angle of measure 32.9977o are
similar by the ______ criterion. Therefore, we know that the values of
corresponding ratios of side lengths will be equal.
Let's Sum It Up!!
• The sine of one acute angle of a right triangle is equal to the cosine of the
other acute angle in the triangle.
• The tangent of one acute angle of a right triangle is equal to the reciprocal of
the tangent of the other acute angle in the triangle.
• The ratios we write for the sine, cosine, and tangent of an angle are useful
because they allow us to solve for two sides of a triangle when we know only
the length of one side.
Name_____________________
Date _____________________
CC Geometry H
HW #23
Leave all answers in simplest radical form and rationalize denominators when necessary.
1) Given the triangle in the diagram, complete the following table.
2) Given the table of values below (not in simplest radical form), label the sides and
angles in the right triangle
3) a. Given the diagram of the triangle, complete the table.
b. Which ratios are equal?
c. How are tan s and tan t related?
4) Given sin and sin β, complete the missing values in the table. You may draw a
diagram to help you.
5) Given the triangle below, fill in the missing values in the table.
6) Jules thinks that if
and β are two different acute angle measures, then
sin ≠ sinβ. Do you agree or disagree?
Review:
1) The following four right triangles are similar. Find the unknown side lengths for
each triangle.
60
60
6
30
30
2) Find the area of ΔSUN in simplest radical form and find the measure of SN in
simplest radical form.
S
U
N