Download 10.1A Right Triangle Trigonometry

Chapter 10 Trigonometric Functions
10.1A Right-Angle Trigonometry
Objectives:
F.TF.3: Use special triangles to determine geometrically the values of sine, cosine, and tangent
for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and
tangent for π – x, π + x, and 2π – x in terms of their values for x, where x is any real number.
F.TF.5: Choose trigonometric functions to model periodic phenomena with specified amplitude,
frequency, and midline.
For the board: You will be able to understand and use trigonometric relationships of acute angles in a
triangle and determine side lengths of right triangles by using trigonometric functions.
Anticipatory Set:
Geometry Review:
Two triangles are similar if 2 angles of one triangle are congruent to 2 angles of another.
This means that all right triangles with a 40° angle are similar to all other right triangles with a 40° angle.
E
ΔGBA ~ ΔFCA ~ ΔEDA
Similar triangles have proportional sides.
F
GB FC ED
GB FC ED
AB AC AD
and
and






G
AB AC AD
AG AF AE
AG AF AE
Each of these ratios is associated with m<A = 40°.
40°
A
B
C
D
A trigonometric function is a function whose rule is given by a trigonometric ratio.
A trigonometric ratio compares the lengths of two sides of a right triangle.
The Greek letter θ is traditionally used to represent the measure of an acute angle in a right triangle.
The values of the ratios depend upon θ and these ratios are usually described using the words adjacent,
opposite and hypotenuse.
B
BC is opposite <A
AC is adjacent to <A
AB is the hypotenuse
(the hypotenuse is always across from the right angle)
C
Instruction:
You may remember SohCahToa.
A
Trigonometric Functions
Words
The sine (sin) of angle θ is the ratio of the length
of the opposite leg to the length of the
hypotenuse
The cosine (cos) of angle θ is the ratio of the
length of the adjacent leg to the length of the
hypotenuse
The tangent (tan) of angle θ is the ratio of the
length of the opposite leg to the length of the
adjacent leg.
Numbers
sin θ =
Symbols
4
5
3
cos θ =
5
sin θ =
4
5
opposite
hypotenuse
cos θ =
adjacent
hypotenuse
tan θ =
opposite
adjacent
θ
3
tan θ =
4
3
Open the book to page 693 and read example 1.
Example: Find the value of the sine, cosine, and tangent functions for θ.
24
θ
sin θ =
25
25
7
7
cos θ =
34
24
tan θ =
24
7
White Board Activity:
Practice: Find the value of the sine, cosine, and tangent functions for θ.
θ
34
16
30
30 15

34 17
16 8
cos θ =

34 17
30 15
tan θ =

16 8
sin θ =
The reciprocals of the sine, cosine, and tangent functions are also trigonometric functions.
They are the trigonometric functions cosecant, secant, and cotangent.
Reciprocal Trigonometric Functions
Words
The cosecant (csc) of angle θ is the reciprocal of
the sine function.
The secant (sec) of angle θ is the reciprocal of
the cosine function.
Numbers
csc θ =
5
4
5
sec θ =
3
The cotangent (cot) of angle θ is the reciprocal
of the tangent function.
Symbols
csc θ =
4
5
hypotenuse
opposite
sec θ =
hypotenuse
adjacent
cot θ =
adjacent
opposite
θ
3
cot θ =
3
4
Open the book to page and read example 5.
θ
Example: Find the values of the six trigonometric functions.
x2 + 242 = 742
x2 = 4900
x = 70
sin θ = 24/74 = 12/37
csc θ = 37/12
cos θ = 70/74 = 35/37
sec θ = 37/35
tan θ = 24/70 = 12/35
cot θ = 35/12
74
24
White Board Activity: Find the values of the six trigonometric functions. θ
182 + 802 = x2
x2 = 6724
x = 82
18
sin θ = 80/82 = 40/41
csc θ = 41/40
cos θ = 18/82 = 9/41
sec θ = 41/9
tan θ = 80/18 = 40/9
cot θ = 9/40
80
Assessment:
Question student pairs.
Independent Practice:
Text: pg. 697 - 698 prob. 2 – 4, 10 – 15, 21 – 23.
For a Grade:
Text: pg. 697 – 698 prob. 4, 14, 22.