algebra & trigonometry by cynthia young - weebly
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PreCalculus AlgebraCynthia Young
Chapter 4.2 Right Triangle Trigonometry
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Chapter 4 Section 2 Right Angle Trigonometry
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Isosceles Right Triangle: a right triangle with two sides that
are congruent. 1) Find the measures of the base angles.
x x
h
y⁰ y⁰
𝑦∘ + 𝑦∘ + 90 = 180
2𝑦∘ = 90 𝑦 = 45∘
x x
h
45⁰ 45⁰
2) “X” can be any number. To make it really easy, lets just
make x = 1.
h
45⁰ 45⁰
1 1
3) Solve for ‘h’.
222 cba 222 11 c
22 c
2c
45⁰ 45⁰
1 1
2
“One-One-Two-Root”
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Use scale factors or proportions to solve for the lengths of sides
of similar 45-45-90 right triangles.
45
45
x1
12
45
45
Write a proportion (equation where
a fraction equals a fraction)
3 14
2=𝑥
1
143x
3 2 7
2= 𝑥
𝑥 = 3 7
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Use scale factors or proportions to solve for the lengths of
sides of similar 45-45-90 right triangles.
45
45
x1
12
45
45
Write a proportion (equation where a fraction equals a fraction)
4 10
2=𝑥
1x
4 2 5
2= 𝑥
𝑥 = 4 5
4 10
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60º
2
60º
30º
2
2
1 1
30º
By constructing angle bisector of the top angle of a 60-60-60
equilateral triangle, two triangles are formed.
Corresponding parts of congruent triangles are
congruent (CPCTC) (all remaining corresponding
pairs of angles and sides are congruent).
Are the two triangles congruent? Why? ASA
Length = 1 and length = 1
Bottom legs (of the right
triangles) are congruent so each
is ½ the total of the original
triangle’s bottom length).
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2x
1
30
60
We now have a 30-60-90 triangle.
Solve for ‘x’.222 cba
𝑥2 + 12 = 22
𝑥2 = 4 − 1
𝑥2 = 3𝑥 = 3
“one-two-three-root”2
1
30
603
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30-60-90 Right Triangle
60
30
1
Solve with a proportion
Write a proportion (equation where a fraction equals a
fraction)
60
30
X
y
6 2
2=𝑥
1
2 ∗ 3 ∗ 2
2=𝑥
1
6 2 2 3
3 2 = 𝑥
6 2
2=
𝑦
3
2 ∗ 3 ∗ 2
2
3 2 =𝑦
3
3 2 =𝑦
3
⇒
⇒
⇒
3 6 = 𝑦⇒
⇒ ⇒
∗ 3 ∗3
1
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Special Triangles
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Solving Right Triangles
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Solving Right Triangles given two sides
𝑠𝑖𝑛𝛽 =𝑜𝑝𝑝
ℎ𝑦𝑝=19.67 𝑐𝑚
37.21 𝑐𝑚
𝑠𝑖𝑛𝛽 = 0.5286
sin−1( 𝑠𝑖𝑛𝛽) = sin−1(0.5286)
𝛽 = 37.9
𝛼 = 180 − 𝛽
𝛼 = 148.1
𝑐𝑜𝑠𝛽 =𝑎𝑑𝑗
ℎ𝑦𝑝
𝑐𝑜𝑠37.9 =𝑎
37.21 𝑐𝑚
0.7891 =𝑎
37.21 𝑐𝑚
𝑎 = 29.4 𝑐𝑚
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Using Bearing
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-Notice that the bearing (direction the plane is pointed) is relative to
north. We can assume that the position of the control tower is at the
location where the plane took off.
-We can draw a right triangle with leg lengths 5 and 12 miles since the
plane turned 90 degrees to the left between the initial bearing of 28 N and
the next bearing.
-We can calculate the bottom angle of the right triangle using the tangent
ratio.𝜃 = tan−1
12
5= 67.4
To find the plane’s bearing from the control tower,
we must subtract the plane’s initial bearing from
theta. 𝛽 = 67.4 − 28 𝛽 = 39.4We can describe the bearing as 39.4 degrees to
the west of north: (N 39.4 W) or using the true
direction it is 39.4 degrees to the left of north
(39.4 less than 360) or 320.6 “true.”