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4.2a: Right Triangle Trigonometry
p. 412-419GSE’s Covered
Primary: M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving angles,lines, polygons, circles, or right triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts (e.g., Pythagorean Theorem, Triangle Inequality Theorem).
Secondary: M(G&M)–10–9 Solves problems on and off the coordinate plane involving distance, midpoint, perpendicular and parallel lines, or slope.
Using the reference angle for the right triangles above, identify: adjacent side, opposite side, hypotenuse.
Reference angle- an acute angle used in the right triangle
SOHCAHTOA
hyp
oppref
angle) sin(
All are sides of right triangles
hyp
aref
djangle) cos(
adj
oref
ppangle) tan(
Replace thisWith either the angleOr variable
What does it mean?
hyp
opprefangle )sin(
The sine of the reference angle is the ratio of the opposite side to the hypotenuse of a right triangle.
x
The angle we are talking about
The opposite side to the angle we are talking about
Always the hypotenuse in a right triangle
8 in9 in
So, sin x = 9
8
Lets solve this equation
x
A B
C
4 in
10 in
hyp
oppx
sin
10
4sin x
To solve for the angle, we need to get rid of sin
To get rid of sin and solve for the angle we use on both sides
1sin
10
4)(sinsin)(sin 11 x
10
4sin x
10
4sin 1x
24x Which means the angle is about 24 degrees
50
6 in x
Solve for x
Label the information you have in the triangle
Reference angle
AdjacentSide to The refangle
hypotenuse
If we have the Adjacent side and the Hypotenuse, think SOHCAHTOA
hyp
ax
djcos
x
650cos Now solve
For x
)( 6
50cos)( xx
x Multiple both sides by x
650cos)( x
50cos
6)( x Divide both sides by
Cos 50
inx 33.9Which means the hypotenuse is 9.3 in
70
8 ft
X ft
Solve for x
Label the information you have in the triangle
If we have the Opposite side and the Adjacent, think SOHCAHTOA
Adjacent side to the ref angle
Opposite side to the ref angle
adj
ox
pptan
8
70tan
x
)8(8
70tan)8(
x Multiply both sides by 8
x70tan)8(You have x alone, so evaluate 8 tan 70
ftx 22 So the opposite side is approximately 22 ft
Example on the coordinate plane
A (8,2)
B (4,5)
C (7,9) ABCin Cm Find
Secondary: M(G&M)–10–9 Solves problems on and off the coordinate plane involving distance, midpoint, perpendicular and parallel lines, or slope.
Primary: M(G&M)–10–2
Phil stands on the sidewalk of a road. Phil’s favorite pizza restaurant is on the other side of the road. His estimated line of sight to the pizza place is 43 degrees. He needs to go to the post office at some point which is 120 feet up the road he is standing on. The line of sight from the post office to the pizza place is 90 degrees.
How far of walk would it be for Phil from his original position to the pizza place?
How far is the walk from the post office to the pizza place?