Download Today`s Goal: Apply the tangent ratio to solve problems

Mathematics 10
Kim
3.1 The Tangent Ratio
Today’s Goal: Apply the tangent ratio to solve problems involving right
triangles.
Let’s start with some activity!
1. Use a protractor and a ruler to build 3 right triangles of different sizes.
The measure of one of the 3 angles should be 30 degrees.
2. Label one of the 3 triangles in order to show the leg that is adjacent to
the angle of 60 degrees and the leg that is opposite to the angle of 60
degrees.
For example,
Draw your 3 triangles here:
Mathematics 10
Kim
3. Measure the legs of all 3 triangles as accurate as possible. (opposite and
adjacent legs only!) Record each length in the table below.
Triangle #1
Triangle #2
Triangle #3
leg opposite to 60°
leg adjacent to 60°
Ratio of
𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 60°
𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 60°
4. Repeat the activity with angle of 45 degrees.
Your 3 triangles here:
Triangle #1
leg opposite to 45°
leg adjacent to 45°
Ratio of
𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 45°
𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 45°
5. What have you discovered?
Triangle #2
Triangle #3
Mathematics 10
Kim
This specific ratio, also called trigonometric ratio, is called tangent ratio.
In general, tangent of angle A =
𝐥𝐞𝐧𝐠𝐭𝐡 𝐨𝐟 𝐥𝐞𝐠 𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞 𝐚𝐧𝐠𝐥𝐞 𝐀
𝐥𝐞𝐧𝐠𝐭𝐡 𝐨𝐟 𝐥𝐞𝐠 𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭 𝐭𝐨 𝐀
tan(A) =
𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞
𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭
=
𝒂
𝒃
Let’s see how it looks like in examples:
Ex. 1
Write each trigonometric ratio. Leave your answer in a reduced fraction.
a) tan A
B
20
12
b) tan B
A
C
16
Your Turn
Calculate each trigonometric ratio.
a) tan L
5
N
M
12
13
b) tan N
L
Mathematics 10
Kim
Ex. 2 Using the TAN ratio to determine the Measure of an Angle:
a) Calculate tan 25o to four decimal places?
b) Draw a triangle to represent
tenth of a degree.
5
tan 𝜃 = .
4
Calculate the angle 𝜃 to the nearest
Your Turn:
a) Calculate tan 30o to four decimal places.
b) Determine the measures of ∠𝐺 and ∠𝐽 to the nearest tenth of a degree.
Mathematics 10
Kim
Tangent ratio often can help us measure immeasurable distances and
heights.
Let’s learn a definition and try a real-life example.
Definition:
Angle of Inclination – This is the ACUTE angle that a line makes with the horizontal
Ex. 3
A surveyor wants to determine the width of a river for a proposed
bridge. The distance from the surveyor to the proposed bridge site is
400 m. The surveyor uses a theodolite to measure angles. The
surveyor measures a 31 o angle to the bridge site across the river.
What is the width of the river, to the nearest metre?
HW: WS
TEXTBOOK
Finish Trigonometry on Tangent Ratio - on a separate piece of paper.
pg.107-109
# 1-6