Trigonometry Sine SOH Cosine CAH Tangent TOA Inverse

Trigonometry

Sine S–O–H

Cosine C–A–H

Tangent T–O–A

Inverse Sine Arcsin (x) or -1 Sin (x) -1 Sin (a/c)=α⁰

Inverse Cosine Arccos (x) or -1 Cos (x) -1 Cos (b/c)=α⁰

Inverse Tangent Arctan (x) or -1 Tan (x) -1 Tan (a/b)=α⁰

Solving Right Triangles Given two sides. 1) Use the Pythagorean Theorem to find 3 rd side. 2) Use preferred inverse trig ratio to find angles.

Solving Right Triangles Given one side and one angle. 1) Use preferred trig ratio to find 2 nd side. 2) Use the Pythagorean Theorem to find 3 rd side. 3) Use Triangle Sum Theorem to find 2 nd angle.

Similar Right Triangles Small Triangle Separate the triangles! Medium Triangle Large Triangle

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

Example: Given Sides The triangle isthe similar to 2 the medium Now complete large triangle. Thesmall Which triangle is has now complete. quantities? Large Medium Small Triangle triangle. 15 + b = 25 9 +12 =a 2 22 theb 2 triangles? What do you 2 know about =16 2 other

Example: Given Angle & Side. Remember, when you know one angle, you can find Complete Finally, complete the medium the large triangle. side. Which triangle has the side? the others. 10√ 3 10 20

The Shadow Problem The Washington Monument casts tree casts a shadow… flagpole casts a shadow… a. Ashadow… The Washington If the flagpole is shadow is. Monument 285’ 35’ tall long isand 169 m tall. If the shadow the sun’s rays makes isan 100 m long, what angle of 55⁰ of 40⁰ with the does the sun’s rays make the flagpole, tree, how tallhow is the long tree? is with monument? Round the Round shadow? to the Round nearestto foot. the th. tonearest the nearest 100 th 10.

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

Angles of Elevation The 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 10 th.

Angles of Depression The 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? No Use tangent. If you want the angle, use inverse tangent. Yes Which do you have/want – opp or adj? Opp Adj Use Cosine. If you want the angle use inverse cosine. Use Sine. If you want the angle use inverse sine.