Download Lesson 13-4 Special Right Triangles and Exact Values

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Algebra 2: Lesson 13-4 Special Right Triangles and Exact Value
Learning Goals:
1) What are the special right triangles and how do we use them to find the exact value of the
trigonometric functions for angles in the first quadrant?
2) What is the co-function relationship and how do we use it?
Special Right Triangles


To determine the sine, cosine, and tangent values at 30o, 45o, and 60o, there are 2 special right
triangles that we can get the values from.
Below are the triangles where these trig values come from.
45 – 45 – 90
30 – 60 – 90
Warm – Up
Use the triangles above and what you already know about right triangle trig, to find the trig values
listed in the chart below. Be sure to rationalize all denominators.
This chart must be memorized!!
Determine the exact value of each of the following expressions in simplest radical form:
1) sin 30o + cos 30o
2) (csc 45o)(tan 60o)
3)  cos 60o + sin 60o - cot 60o
4) sec 30o + tan 30o
5) If
√
and
is in quadrant I, find
.
6) If
√
and
is in quadrant I, find
.
Let’s explore some other relationships…
Answer the following questions with a partner.

What type of triangle is shown here? How do you know?

Using your answer from Question # 1 and your knowledge of the
interior angles of triangles, what MUST be the sum of
Therefore the two non-right angles in a right triangle must always be

Write the indicated trig ratios using the sides of the triangle shown above :
=
a) What do you notice about these values?
b) Will this be true for ALL right triangles? Try another!
=
Summarize your findings with your group members:
________.
Let’s summarize as a class:
The VALUE of sine of an acute angle is equal to the _
Symbols:
_________of its complement.
Example:
The VALUE of cosine of an acute angle is equal to the
complement.
Symbols:
of its
Example:
VALUES are equal, ANGLES are complements
We call Sine and Cosine
.
Practice!
1) Write each expression as a function of an acute angle whose measure is less than 45˚
a. sin 80˚
b. cos 36˚
2) Each equation contains the measures of two acute angles. Find a value of θ for which the
statement is true.
a) sin 10° = cos θ
b) sin θ = cos 2θ
3) In right triangle ABC with the right angle at C,
and state the value of x. Explain your answer.
and
. Determine
4) Express sin 390o in terms of its cofunction.
5) If x in an acute angle and sin x = 3/5, then
(1) cos x = 3/5
(2) cos x = 2/5
(3) cos (90 – x) = 2/5
(4) cos (90 – x) = 3/5
6) Suppose
sin( )
represents a number of degrees of rotation. Is it possible that
? Explain.
7) Given that the cos (90 – x) = 4/5, find cot x.
( )
and