Download Lesson 12: Defining Trigonometric Ratios

Lesson 12
RCSD Geometry Local MATHEMATICS CURRICULUM
Name____________________________
U8
Date_____________ Period _________
Lesson 12: Defining Trigonometric Ratios
Learning Targets

For a given acute angle of a right triangle, I can identify the opposite side, adjacent side, and hypotenuse.

Given the side lengths of a right triangle with acute angles, I can find sine, cosine, and tangent of each
acute angle
Opening Exercise (4 minutes) Use the right triangle ∆𝐴𝐵𝐶 to answer 1–3.
1. Name the side of the triangle opposite ∠𝐴.
2. Name the side of the triangle opposite ∠𝐵.
3. Name the side of the triangle opposite ∠𝐶.
New concept: The triangle below is right triangle ∆𝐴𝐵𝐶. Denote ∠𝐵 as the right angle of the triangle.

Mark ∠𝐴 in the triangle with a single arc.
With respect to acute angle ∠𝐴 of a right triangle △ 𝐴𝐵𝐶, we identify the opposite
side, the adjacent side, and the hypotenuse as follows:

̅̅̅̅ ,
With respect to ∠𝐴, the opposite side, denoted 𝑜𝑝𝑝, is side 𝐵𝐶
With respect to ∠𝐴, the adjacent side, denoted 𝑎𝑑𝑗, is side ̅̅̅̅
𝐴𝐵 ,

The hypotenuse, denoted ℎ𝑦𝑝, is side ̅̅̅̅
𝐴𝐶 and is opposite from the 90˚ angle.

Example 1
For each exercise, label the appropriate sides as adjacent, opposite, and hypotenuse, with respect to
the marked acute angle.
a.
b.
c.
Lesson 12
RCSD Geometry Local MATHEMATICS CURRICULUM
Name____________________________
Date_____________ Period _________
Example 2
1. Identify the
2. Identify the
3. Identify the
𝒐𝒑𝒑
𝒉𝒚𝒑
𝒂𝒅𝒋
𝒉𝒚𝒑
𝒐𝒑𝒑
𝒂𝒅𝒋
ratio for angles ∠𝑨
ratio for angles ∠𝑨.
ratio for angles ∠𝑨
In any right triangle, the ratios found above are called trigonometric ratios
To remember the Basic Trigonometric Functions we use (SOH – CAH – TOA)
Example 3. In ∆𝑷𝑸𝑹, 𝒎∠𝑷 = 𝟓𝟑. 𝟐° and 𝒎∠𝑸 = 𝟑𝟔. 𝟖°. Complete the following table.
Measure
of Angle
𝟓𝟑. 𝟐
𝟑𝟔. 𝟖
𝒐𝒑𝒑
𝑺𝒊𝒏𝒆 𝜽 = (
)
𝒉𝒚𝒑
U8
𝒂𝒅𝒋
𝑪𝒐𝒔𝒊𝒏𝒆 𝜽 = (
)
𝒉𝒚𝒑
𝑻𝒂𝒏𝒈𝒆𝒏𝒕 𝜽
𝒐𝒑𝒑
=(
)
𝒂𝒅𝒋
Lesson 12
RCSD Geometry Local MATHEMATICS CURRICULUM
Name____________________________
U8
Date_____________ Period _________
Now you try
Example 4 . In the triangle below, 𝒎∠𝑨 = 𝟑𝟑. 𝟕° and 𝒎∠𝑩 = 𝟓𝟔. 𝟑°. Complete the following table.
Measure of Angle
Sine
Cosine
Tangent
𝟑𝟑. 𝟕
𝟓𝟔. 𝟑
Example 5 . In the triangle below, let 𝒆 be the measure of ∠𝑬 and 𝒅 be the measure of ∠𝑫. Complete the
following table.
Measure of Angle
Sine
Cosine
𝒅
𝒆
Closing (3 minutes)
Describe the ratios that we used to calculate sine, cosine, and tangent.
a. Given an angle, 𝜃, sin 𝜃 =
, cos 𝜃 =
, and tan 𝜃 =
Tangent
Name____________________________
U8
Lesson 12
RCSD Geometry Local MATHEMATICS CURRICULUM
Date_____________ Period _________
Lesson 12: Defining Trigonometric Ratios
Classwork
1. In the triangle below, let 𝒙 be the measure of ∠𝑿 and 𝒚 be the measure of ∠𝒀. Complete the following
table.
Measure of Angle
Sine
Cosine
Tangent
𝒙
𝒚
2. Given the diagram of the triangle, complete the following table.
Angle Measure
𝐬𝐢𝐧 𝜽
𝐜𝐨𝐬 𝜽
𝐭𝐚𝐧 𝜽
𝒔
𝒕
a. Which values are equal?
b. How are 𝐭𝐚𝐧(𝒔) and 𝐭𝐚𝐧(𝐭) related?
2
3. If 𝑢 and 𝑣 are the measures of complementary angles such that sin 𝑢 = 5 and tan 𝑣 =
and angles of the right triangle in the diagram below with possible side lengths.
√21
,
2
label the sides
Lesson 12
RCSD Geometry Local MATHEMATICS CURRICULUM
Name____________________________
U8
Date_____________ Period _________
Lesson 12: Defining Trigonometric Ratios
Homework
1. Given the triangle in the diagram, complete the following table.
𝐬𝐢𝐧
Angle Measure
𝐜𝐨𝐬
𝐭𝐚𝐧
𝛼
𝛽
2. Given the table of values below (not in simplest radical form), label the sides and angles in the right triangle
Angle Measure
𝛼
𝛽
sin
4
cos
2√6
tan
4
2√10
2√6
2√10
4
2√10
2√10
2√6
2√6
4
3. Given sin 𝛼 and sin 𝛽, complete the missing values in the table. You may draw a diagram to help you.
Angle Measure
𝛼
sin
√2
cos
5
3√3
3√3
tan
𝛽
********4. Given the triangle shown to the right, fill in the missing values in the table.
Hint: Use the Pythagorean Theorem to find hypotenuse:
Angle Measure
𝛼
𝛽
𝐬𝐢𝐧
𝐜𝐨𝐬
𝐭𝐚𝐧