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UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
Common Core Standards
SRT.C.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle,
leading to definitions of trigonometric ratios for acute angles
SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles
SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems
SRT.D.10: Prove the Laws of Sines and Cosines and use them to solve problems
SRT.D.11: Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in
right and non-right triangles
Vocabulary
Right Triangle
Tangent
Trigonometric Function
Law of Sines
30-60-90 Triangle
Law of Cosines
45-45-90 Triangle
Unit Circle
Sine
Cosine
Prerequisite Key Concepts
The sum of the three interior angles in a triangle is
two angles add to
.
. If one of those angles is 90o, then the other
Pythagorean Theorem can only be applied to
triangles.
Any two triangles, with the same angles, are similar by
This means that their side lengths are
(theorem/postulate).
.
In order to make a unique triangle, you need to know at least _______ parts of the triangle.
What orders of parts will always make a unique triangle?
AAA
SSS
ASA
AAS
SAS
ASS
ASA
AAS
SAS
ASS
AAS
SAS
ASS
What orders of parts will sometimes?
AAA
SSS
What orders of parts will never make a unique triangle?
AAA
SSS
ASA
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
Lesson 1 – Special Right Triangles


I can apply the 30-60-90 right triangle relationships.
I can apply the 45-45-90 right triangle relationships.
We are going to establish the relationships in a 45-45-90 triangle and a 30-60-90 triangle.
45-45-90 Triangle:
B
45°
45°
A
C
30-60-90 Triangle:
B
30°
60°
A
C
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
For each triangle, fill in the missing sides and angles. Your answers must be in exact form and rounded
to the nearest tenth.
B
A
45°
5°
45°
45°
C
A
B
C
135
A
-90
C
B
45°
B
C
45°
45°
A
B
135
E
C
A
-90
B
B
60°
30°
30°
60°
120
C
A
30°
30°
60°
60°
A
C
B
30°
C
60°
C
A
30°
B
B
60°
30°
A
60°
120
D
A
C
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
Lesson 2 – The Trigonometric Functions



I can state the trig ratios of a right triangle.
I can explain why any right triangle yields the same trig values.
I can explain the relationship of sine and cosine with complementary angles
Any two right triangles, with one other angle congruent, are similar by AA Similarity.
This means that their side lengths are
.
These ratios between the side lengths are trigonometric ratios.
In a right triangle,
The sine of an angle is the ratio
The cosine of an angle is the ratio
The tangent of an angle is the ratio
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
Ex. Write each trigonometric ratio for angle J and angle K
Ex. Find an approximate value of tan72 without typing it in a calculator. Show work below.
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
We now know that the trigonometric ratios are not dependent on the sides, but the ratios. Therefore,
there is one fixed value for every angle, from 0 to 90.
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
Lesson 3 – Finding Sides of Right Triangles


I can solve right triangles using trig functions.
I can solve right triangles using inverse trig functions.
We learned yesterday that these are the trigonometric ratios.
One application of the trigonometric ratios is to use them to find the missing sides of a right triangle.
To find a side, using the trig ratios:
1. Pick an acute angle in the triangle.
2. Label the sides as opposite, adjacent, and hypotenuse in relation to that angle.
3. Using the definitions, write proportions using the angle picked and the sides.
5. Solve the necessary proportion.
Solve the triangle. Round each side to one decimal place.
Caution!
Be sure your
calculator is in
degree mode, not
radian mode.
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
Lesson 4 – Finding Angles of Right Triangles


I can solve right triangles using trig functions.
I can solve right triangles using inverse trig functions.
Recall the trigonometric ratios.
What happens if you have the sides of the right triangle, but don’t have the angles?
The inverse trig functions are used to find the angle, with a given ratio.
Another application of the trigonometric ratios is to use them to find the missing angles of a right
triangle. All you need in the right angle is two side lengths.
To find an angle, using the trig ratios:
1. Pick an acute angle in the triangle to find.
2. Label the sides as opposite, adjacent, and hypotenuse in relation to that angle.
3. Write a ratio with two known sides.
4. Use the inverse function to find the angle.
Solve the triangle. Round each angle to the nearest tenth of a degree.
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
Lesson 5 – Angle of Elevation and Depression


I can use angles of elevation to find desired measurements.
I can use angles of depression to find desired measurements.
An angle of elevation and angle of depression are formed with the
_
____.
An inquisitive math student is standing approximately 308 feet from the base of the Washington
Monument. The angle of elevation from the ground to the top of the monument is 61  . How tall is the
Washington Monument?
Suppose a forest ranger in a 90-foot observation tower sees a fire and the angle of depression to the
fire is 3°. What is the horizontal distance to this fire? Round to the nearest foot.
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
Lesson 8 – The Unit Circle

I can state the cosine, sine ordered pairs on the unit circle.
The unit circle is a circle with a radius of one unit, centered at the origin.
We can represent any angle with its vertex centered at the origin and initial side on the positive x-axis
on the unit circle.
The purpose is to extend our definitions of the trigonometric functions to other angles, other than acute
angles.
Conclusions
The x-coordinate of a point on the unit circle is equivalent to the
.
The y-coordinate of a point on the unit circle is equivalent to the
.
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
Lesson 6 – Using the Law of Sines

I can solve triangles using the Law of Sines.
As we’ve discussed before, you may only use SOHCAHTOA on right triangles.
To find parts of non-right triangles, the law of sines and law of cosines can be used.
Law of Sines
Remember, to solve any triangle, you must know
triangle without knowing at least one side length.
parts of the triangle. You cannot solve a
Use the Law of Sines to find the missing parts. Round side lengths and angles to the nearest tenth.
Solve the triangle.
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
Solve the triangle.
Solve the triangle.
Solve the triangle.
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
Lesson 7 – Using the Law of Cosines

I can solve triangles using the Law of Cosines.
To solve any triangle, you must know
parts of the triangle.
Look at the two triangles. Why can’t Law of Sines be used find a missing part with the given
information?
These triangles require the use of the Law of Cosines to solve them.
Law of Cosines
Find the missing parts. Round side lengths and round angles to the nearest tenth.
Solve the triangle.
UNIT 6: RIGHT TRIANGLES AND TRIGONOMETRY
Solve the triangle.
Solve the triangle.
Solve the triangle.