Trigonometry

SineS – O – H

CosineC – A – H

TangentT – O – A

a⁰ + b⁰ = 90⁰

a⁰ + b⁰ = 90⁰

a⁰ + b⁰ = 90⁰

Inverse SineArcsin (x) or Sin-1(x)

Sin-1(a/c)=α⁰

Inverse CosineArccos (x) or Cos-1(x)

Cos-1(b/c)=α⁰

Inverse TangentArctan (x) or Tan-1(x)

Tan-1(a/b)=α⁰

Solving Right TrianglesGiven two sides.

1) Use the Pythagorean Theorem to find 3rd side.

2) Use preferred inverse trig ratio to find angles.

Solving Right TrianglesGiven one side and one angle.

1) Use preferred trig ratio to find 2nd side.

3) Use Triangle Sum Theorem to find 2nd angle.

2) Use the Pythagorean Theorem to find 3rd side.

Similar Right TrianglesSeparate the triangles!

Small Triangle

Medium TriangleLarge Triangle

What are the angles?Remember, a + b = 90°

Example: Given SidesWhich triangle has 2 quantities?

92+122=a2

The small triangle is now complete.

What do you know about the other triangles?

The small triangle is similar to the medium triangle.

b =16

Now complete the large triangle.

152 + b2 = 252

Small TriangleMedium TriangleLarge Triangle

Example: Given Angle & Side.Remember, when you know one angle, you can find the others.Which triangle has the side?Complete the medium triangle.Finally, complete the large side.

1010√3

20

The Shadow ProblemA tree casts a shadow…A flagpole casts a shadow…The Washington Monument casts a shadow…

If the flagpole is 35’ tall and the sun’s rays makes an angle of of 40⁰ with the flagpole, how long is the shadow? Round to the nearest 10th.

If the shadow is 285’ long and the sun’s rays makes an angle of 55⁰ with the tree, how tall is the tree? Round to the nearest foot.

The Washington Monument is 169m tall. If the shadow is 100m long, what angle does the sun’s rays make with the monument? Round to the nearest 100th.

Angles of Elevation/DepressionWhich one you use depends on your point of view. However, they are essentially the same thing. Why?

Angles of ElevationThe Ramp Problem

The smaller the angle, the easier it is to push something up the ramp. If the bed of a truck is 3.5’ high and you want a ramp with a 10⁰ angle, how long (along the diagonal) does the ramp have to be? Round to the 10th.

Angles of DepressionThe Lighthouse Problem

A lighthouse keeper is looking at a boat at sea. If his line of sight is 20⁰ from the horizontal

and the light is 95’ from sea level, how far is the boat from shore? Round to the nearest foot.

Angles of ????The Ski Slope Problem A downhill racer skis a

mountain with a 28⁰ grade (against the horizontal). If the vertical height of the slope is 4,500‘ and he completed the run in 2.5 minutes, how fast did he go? Find the answer in feet/minute, then mph. Round to the nearest mile.

Angles of ????The Kite/Balloon/Plane Problem

During the Civil War, the Union Army used hot air balloons for reconnaissance. If the balloonist is 1000’ high and he sees cannons

when the angle of his line of sight is 12⁰, how far are the cannons from his base? Round to the nearest foot.

Which to use? Sin or Cos or Tan?

Do you have or do you want the

hypotenuse?

Use tangent. If you want the angle,

use inverse tangent.

Which do you have/want – opp or adj?

Use Sine. If you want the angle

use inverse sine.

Use Cosine. If you want the angle

use inverse cosine.

No

YesOpp

Adj