Trig Ratios and Cofunction Relationships

Trig Ratios SOH-CAH-TOA

SINE Pronounced “sign”

Pronounced “co-sign” COSINE

Pronounced “tan-gent” TANGENT

Pronounced “theta” Greek Letter  Represents an unknown angle

opposite hypotenuse adjacent hypotenuse opposite adjacent

Finding sin, cos, and tan. Just writing a ratio.

1. Find the sine, the cosine, and the tangent of theta. Give a fraction Shrink yourself down and stand where the angle is. Identify your hypotenuse, adjacent side, and opposite side. H A O

2. Find the sine, the cosine, and the tangent of theta Shrink yourself down and stand where the angle is. Identify your hypotenuse, adjacent side, and opposite side. H A O

Sin-Cosine Cofunction

The Sin-Cosine Cofunction

7. Sin 28 = ?

8. Cos 10 = ?

What is Sin Z? What is Cos X?

What is sin A? What is Cos C?

9.  ABC where  B = 90. Cos A = 3/5 What is Sin C?

10. Sin  = Cos 15 What is  ?

Draw  ABC where  BAC = 90  and sin B = 3/5 11. What is the length of AB? 12. What is tan C? 4 4/3

13. Draw stick-man standing where the angle is and mark each given side. Then tell which trig ratio you have. O H sin

A C M If C = 20º, then cos C is equal to: A. sin 70 B. cos 70 C. tan 70

Using Trig to Find Missing Angles and Missing Sides

Finding a missing angle. (Figuring out which ratio to use and an inverse trig button.)

Ex: 1 Figure out which ratio to use. Find x. Round to the nearest tenth. 20 m 40 m Shrink yourself down and stand where the angle is. Identify the given sides as H, O, or A. x A O What trig ratio is this?

Ex: 2Figure out which ratio to use. Find x. Round to the nearest tenth. 15 m 50 m x Shrink yourself down and stand where the angle is. Identify the given sides as H, O, or A. H O What trig ratio is this?

Ex. 3: Find . Round to the nearest degree A O

Ex. 4: Find . Round to the nearest degree H A

Ex. 5: Find . Round to the nearest degree H O

Finding a missing side. (Figuring out which ratio to use and getting to use a trig button.)

Ex: 6Figure out which ratio to use. Find x. Round to the nearest tenth. 20 m x O A

Ex: 7 Find the missing side. Round to the nearest tenth. 80 ft x O A

Ex: 8 Find the missing side. Round to the nearest tenth. 283 m x H O

Ex: 9 Find the missing side. Round to the nearest tenth. 20 ft x A H

When we are trying to find a side we use sin, cos, or tan. When we are trying to find an angle we use ( INVERSE ) sin -1, cos -1, or tan -1.

Trig Application Problems MM2G2c: Solve application problems using the trigonometric ratios.

Depression and Elevation horizontal line of sight horizontal angle of elevation angle of depression

1. Classify each angle as angle of elevation or angle of depression. Angle of Depression Angle of Elevation Angle of Depression Angle of Elevation

Example 2 Over 2 miles (horizontal), a road rises 300 feet (vertical). What is the angle of elevation to the nearest degree? 5280 feet – 1 mile

Example 3 The angle of depression from the top of a tower to a boulder on the ground is 38º. If the tower is 25m high, how far from the base of the tower is the boulder? Round to the nearest whole number.

Example 4 Find the angle of elevation to the top of a tree for an observer who is 31.4 meters from the tree if the observer’s eye is 1.8 meters above the ground and the tree is 23.2 meters tall. Round to the nearest degree.

Example 5 A 75 foot building casts an 82 foot shadow. What is the angle that the sun hits the building? Round to the nearest degree.

Example 6 A boat is sailing and spots a shipwreck 650 feet below the water. A diver jumps from the boat and swims 935 feet to reach the wreck. What is the angle of depression from the boat to the shipwreck, to the nearest degree?

Example 7 A 5ft tall bird watcher is standing 50 feet from the base of a large tree. The person measures the angle of elevation to a bird on top of the tree as 71.5°. How tall is the tree? Round to the tenth.

Example 8 A block slides down a 45  slope for a total of 2.8 meters. What is the change in the height of the block? Round to the nearest tenth.

Example 9 A projectile has an initial horizontal velocity of 5 meters/second and an initial vertical velocity of 3 meters/second upward. At what angle was the projectile fired, to the nearest degree?

Example 10 A construction worker leans his ladder against a building making a 60 o angle with the ground. If his ladder is 20 feet long, how far away is the base of the ladder from the building? Round to the nearest tenth.