Download 45°-90° Special Right Triangle

Section 4.3 Right Angle Trigonometry
Section Objectives: Students will know how to
use the fundamental trigonometric identities.
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Digital Figures, 4–1
The Six Trigonometric Functions
Let A be an acute angle of a right triangle. Then
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Digital Figures, 4–2
45°-45°-90° Special Right Triangle
• In a triangle 45°-45°-90° , the hypotenuse is 2 times as long as
a leg.
Example:
45°
45°
Leg
n
Hypotenuse
5 2 cm
5 cm
n 2
45°
5 cm
u
Leg
45°
Lesson 7-3: Special Right Triangles
3
Trig Functions of Special Angles, 45-45-90
Think of your special triangle rules, what if
your hypotenuse was 1?
sin45=
cos45=
tan45=
30°-60°-90° Special Right Triangle
• In a triangle 30°-60°-90° , the hypotenuse is twice as
long as the shorter leg, and the longer leg is 3 times
as long as the shorter leg.
Example:
Hypotenuse
30°
2n
Longer Leg
n 3
30°
10 cm
5 3 cm
60°
60°
Shorter Leg
u
Lesson 7-3: Special Right Triangles
5 cm
5
Trig Functions of Special Angles, 30-60-90
Think of your special triangle rules, what if
your hypotenuse was 1?
sin30=
cos30=
tan30=
sin60=
cos60=
tan60=
Examples:
Draw a sketch of a right triangle the corresponds
to the following, find the 3rd side, and the 6 trig
functions.
a) 𝑐𝑠𝑐𝜃 =
17
4
b) 𝑡𝑎𝑛𝜃 = 3
Identities
• Reciprocal Identities:
1
𝑠𝑖𝑛𝜃 =
𝑐𝑠𝑐𝜃
1
𝑐𝑜𝑠𝜃 =
𝑠𝑒𝑐𝜃
1
𝑡𝑎𝑛𝜃 =
𝑐𝑜𝑡𝜃
𝑐𝑠𝑐𝜃 =
𝑠𝑒𝑐𝜃 =
𝑐𝑜𝑡𝜃 =
1
𝑠𝑖𝑛𝜃
1
𝑐𝑜𝑠𝜃
1
𝑡𝑎𝑛𝜃
• Quotient Identities:
𝑡𝑎𝑛𝜃 =
𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃
𝑐𝑜𝑡𝜃 =
𝑐𝑜𝑠𝜃
𝑠𝑖𝑛𝜃
Identities
• Pythagorean Identities:
𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1 1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐 2 𝜃
1 + 𝑐𝑜𝑡 2 𝜃 = 𝑐𝑠𝑐 2 𝜃
• Co-function Identities:
𝑠𝑖𝑛𝜃 = cos(90° − 𝜃)
𝑡𝑎𝑛𝜃 = cot(90° − 𝜃)
𝑠𝑒𝑐𝜃 = csc(90° − 𝜃)
𝑐𝑜𝑠𝜃 = sin 90° − 𝜃
𝑐𝑜𝑡𝜃 = tan 90° − 𝜃
𝑐𝑠𝑐𝜃 = sec(90° − 𝜃)
Applications Involving Right Triangles
If the sun is 30° up from the horizon and shining
on a tree forming a 50-foot shadow, how tall is
the tree?
Draw a triangle:
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Digital Figures, 4–10
Example
If a rope tied to the top of a flagpole is 35 feet long,
then what angle is formed by the rope and the ground
when the rope is pulled to the ground, 25 feet from the
base of the pole?
cos θ = 25/35
⇒ θ ≈ 44.42°.
Copyright © Houghton Mifflin Company. All rights reserved.
Digital Figures, 4–11
Verifying Identities
• Get one side = to the other
**Can only work with one side**
Steps:
1. Work with only one side (the complicated
looking side)
2. Look to use your identities, add fractions, get
common denominators, split a fraction, factor,
etc...
3. When in doubt turn everything in terms of sin &
cos (things will usually cancel)
Examples: Verify
1. 𝑠𝑖𝑛𝑥𝑠𝑒𝑐𝑥 = 𝑡𝑎𝑛𝑥
2. tan −𝑥 𝑐𝑜𝑠𝑥 = −𝑠𝑖𝑛𝑥
3. 𝑠𝑒𝑐𝑥 − 𝑠𝑒𝑐𝑥𝑠𝑖𝑛2 𝑥 = 𝑐𝑜𝑠𝑥
4. 𝑐𝑠𝑐𝑥 − 𝑠𝑖𝑛𝑥 = 𝑐𝑜𝑡𝑥𝑐𝑜𝑠𝑥
4.4 Trig Functions of any Angle:
• Let 𝜃 be an angle in standard position with (x,y) a
point on the terminal side of 𝜃 and 𝑟 =
𝑥2 + 𝑦2 ≠ 0
𝑦
𝑠𝑖𝑛𝜃 =
𝑟
𝑥
cos𝜃 =
𝑟
𝑦
tan𝜃 =
𝑥
𝑥≠0
𝑟
𝑐𝑠𝑐𝜃 =
𝑦
𝑟
sec𝜃 =
𝑥
𝑥
𝑐𝑜𝑡𝜃 =
𝑦
𝑦≠0
x≠0
𝑦≠0
Examples:
1. Let (-3,4) be a point on the terminal side of 𝜃.
Find the six trig functions of 𝜃.
2. Let (2,-3) be a point on the terminal side of 𝜃.
Find the six trig functions of 𝜃.
Trig Functions in the Coordinate
Plane
**All Star Trig Class**
There are 2 quadrants
where each function is
positive and to quadrants
where each function
is negative.
Reference Angles
• Let 𝜃 be an angle in standard position. The
Reference Angle is the acute angle 𝜃 ′ formed
by the terminal side of 𝜃 and the horizontal
axis.
QI
QII
𝜃′ = 𝜃
𝜃 ′ = 180° − 𝜃
QIII
𝜃 ′ = 𝜃 − 180°
QIV
𝜃 ′ = 360° − 𝜃
Examples: Find the Reference
Angle 𝜃′
1. 𝜃 = 300°
4. 𝜃 =
5𝜋
6
2. 𝜃 = −135°
5. 𝜃 =
−11𝜋
4
3. 𝜃 = 210°
6. 𝜃 =
−17𝜋
6
Evaluate Each Trig Function using it
reference angle.
1. tan −210° =
3. csc
7𝜋
6
=
2. sin(300°)=
4. cos
23𝜋
4
=
Homework:
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Digital Figures, 4–20
Example: Find the value of a and b.
7 cm
b = 7 2 cm
45°
45°
b
2
x 2
x
45°
45 °
a = 7 cm
a
x
Step 1: Find the missing angle measure.
45°
Step 2: Decide which special right triangle applies. 45°-45°-90°
Step 3: Match the 45°-45°-90° pattern with the problem.
Step 4: From the pattern, we know that x = 7 , a = x, and b = x 2.
Step 5: Solve for a and b
Lesson 7-3: Special Right Triangles
21
Example: Find the value of a and b.
7 cm
b = 14 cm
60°
30°
b
a = 7 3 cm
2x
x 3
60°
30 °
a
x
Step 1: Find the missing angle measure.
30°
Step 2: Decide which special right triangle applies. 30°-60°-90°
Step 3: Match the 30°-60°-90° pattern with the problem.
Step 4: From the pattern, we know that x = 7 , b = 2x, and a = x 3.
Step 5: Solve for a and b
Lesson 7-3: Special Right Triangles
22