Download 8.4 Trigonometry

Warm-Up
Think about what we’ve learned so far in this
chapter. How could you find the value of x in
the triangle below?
8.4 Trigonometry
Trigonometry
• Greek for “triangle measure”
• Uses the angles and sides of a triangle
• Trigonometric ratio is the ratio of 2 side
lengths in a triangle in reference to a specific
angle in the triangle
*The labels for the sides depend on the angle used
*Always use one of the 2 acute angles of the triangle
Hypotenuse Hypotenuse: across from the right angle always (side c)
Side
Adjacent
Side
Opposite
Side
c
Adjacent Side: “next to”
The side adjacent to Angle B is side a
The side adjacent to Angle A is side b
A
Opposite Side: “across from”
The side opposite of Angle B is side b
The side opposite of Angle A is side a
B
a
b
C
Sine Ratio
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
sin 𝐴 =
𝑎
𝑐
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑏
𝑐
𝑏
cos 𝐴 =
𝑐
sin 𝐵 =
Cosine Ratio
Trigonometric
Ratios
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎
𝑐
𝑎
tan 𝐴 =
𝑏
𝑏
tan 𝐵 =
𝑎
cos 𝐵 =
Tangent
Ratio
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
SOH-CAH-TOA
𝒐𝑝𝑝
𝒔𝑖𝑛 =
𝒉𝑦𝑝
𝒂𝑑𝑗
𝒄𝑜𝑠 =
𝒉𝑦𝑝
𝒐𝑝𝑝
𝒕𝑎𝑛 =
𝒂𝑑𝑗
Example 1: Given the triangle, label the sides
(opposite, adjacent, hypotenuse) according to the
specified angle.
a. Angle S
Example 1: Given the triangle, label the sides
(opposite, adjacent, hypotenuse) according to the
specified angle.
b. Angle P
Example 2:
Given the triangle, label the sides (opposite,
adjacent, hypotenuse) according to the
specified angle.
a. Angle R
Example 2:
Given the triangle, label the sides (opposite,
adjacent, hypotenuse) according to the
specified angle.
b. Angle S
Example 1
a) Express sin L as a fraction and as a decimal to
the nearest hundredth.
Find the 3 trigonometric ratios for angle L.
a) sin L
b) cos L
c) tan L
Example 1
d) Express sin N as a fraction and as a decimal to
the nearest hundredth.
Find 3 trigonometric ratios for angle N.
a) sin N
b) cos N
c) tan N
Example 2:
a) Use a special right triangle to
express the cosine of 60° as a
fraction and as a decimal to the
nearest hundredth.
b) Use a special right triangle to
express the tangent of 60° as a
fraction and as a decimal to the
nearest hundreth
Example 3: SOLVE. Round to the
nearest tenth.
a.
x
sin 23 
15
b.
w
cos 74 
5
Example 3: Round to the nearest
tenth.
c.
d.
3
18
sin 80 
tan 60 
x
y
Example 3: Round to the nearest
tenth.
e.
10
cos12 
y
f.
z
tan 68 
43
Example 4:
Use trigonometric ratios to find the value of the
variables. Round to the nearest hundredth.
Example 4:
Use trigonometric ratios to find the value of the
variables. Round to the nearest hundredth.
Change location of 5 and x!
x
27
5
Exit Slip
Find the value of y.
Example 5:
A fitness trainer sets the incline on a treadmill to
7°. The walking surface is 5 feet long.
Approximately how many inches did the trainer
raise the end of the treadmill from the floor?
Example 6:
The bottom of a handicap ramp is 15 feet from
the entrance of a building. If the angle of the
ramp is about 4.8°, about how high does the
ramp rise off the ground to the nearest inch?
Use trigonometric rations to find the value of
the variables. Round to the nearest hundredth
Use trigonometric rations to find the value of
the variables. Round to the nearest hundredth
If sine A = x, then the inverse sine
of x is the measure of angle A.
Inverse Sine
Inverse
Trigonometric
Ratios
*used to find the
measure of an
angle*
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑠𝑖𝑛−1 (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒)
If cos A = x, then the inverse cosine
of x is the measure of angle A.
Inverse Cosine
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑐𝑜𝑠 −1 (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒)
Inverse Tangent
If tan A = x, then the inverse
tangent of x is the measure of angle
A.
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑡𝑎𝑛−1 (𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡)
What is the difference
between Trigonometric
functions and Inverse
Trigonometric
Functions?
• Trigonometric functions (sin, cos, tan) are
used to find side lengths
• Inverse trigonometric functions (sin-1, cos-1,
tan-1) are used to find angle measures
Example 5:
a) Use a calculator to find the measure of P to
the nearest tenth.
Example 5:
b) Use a calculator to find the measure of D to
the nearest tenth.
Example 6:
Solve each right triangle. Round side measures
to the nearest hundredth and angle measures to
the nearest degree.
Example 6:
Solve each right triangle. Round side measures
to the nearest hundredth and angle measures to
the nearest degree. *Change
22°