Download Lesson 1 – Tangent Ratio and its use in calculating lengths

Lesson 1 – Tangent Ratio and its use in calculating lengths
Specific outcome:
- Explain the relationships between similar right triangles and the definitions of the primary
trigonometric ratios (4.1).
- Identify the hypotenuse of a right triangle and the opposite and adjacent sides for a given
acute angle in the triangle (4.2).
Working with Triangles:
-
Vertex is the point where two lines meet.
Each vertex is labelled with an upper case letter.
Each side is labelled with the lower case letter
corresponding to the opposite vertex.
Each side of the triangle is given a special name, relative to angle 𝒙° or ∠ 0:

The longest side is always called the ___________________________________
(label it on the diagram)

The side directly across 𝑥° or ∠O is called the ____________________________
(label it)

The third side is called the ___________________________
(label it).
Example 1: Given the following triangles, identify the name of each side according to
the angle labelled.
a)
b)
c)
Tangent Ratio:
Tangent ∠A =
𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒔𝒊𝒅𝒆 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 ∠𝑨
𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒔𝒊𝒅𝒆 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒕𝒐 ∠𝑨
Note:
𝑎
- A ratio is a fraction
𝑏
-
The Tangent ratio is expressed as a ratio or as a decimal.
Example 2: Determine the tangent ratio of the following. Express each answer as a
ratio.
a) 𝑡𝑎𝑛 𝐴
b) 𝑡𝑎𝑛 𝐵
c) 𝑡𝑎𝑛 𝐵
Example 3: Use your calculator to determine the value of each of the following tangent
ratios to four decimal places.
𝑎) tan 15𝑜
b) tan 60𝑜
c) tan 75𝑜
Note: Use the tangent ratio to determine the measure of an angle, by taking the inverse
of the ratio. ( 𝒕𝒂𝒏−𝟏 , called inverse tan).
Example 4: Calculate the value of the indicated angle, to the nearest degree.
a)
b)
c)
Example 5: Determine the measure of the angle to the nearest degree.
𝑎) tan 𝐴 = 0.8192
b) tan 𝐵 = 0.4226
c) tan 𝐶 =
11
6
Example 6: A 10 ft. ladder leans against the side of a building with its base 6 ft. from
the wall. What angle, to the nearest degree, does the ladder make with the ground?
Practice Questions: Page 75 # 3 – 5(a, c’s), 9(a), 10(c), 14
Steps to determine the side length:
1. Label the sides according to the given angle.
2. Use the tan ratio.
Example 7: Determine the length of the unknown side.
a)
b)
c)
d)
Example 8: The Calgary Tower is 191 m high. At a certain point, the angle between
the ground and the line of sight to the top of the tower is 81o. To the nearest tenth of a
meter, what is the horizontal distance?
Practice Questions: Page 82 # 3 – 5(a, c’s), 6 – 8