7 6 Apply Sine and Cosine Ratios Goal

7. 6 Apply Sine and Cosine Ratios Goal: To use trigonometric ratios for indirect measurements of right triangles.

Trigonometry • Derived from Greek words, “trigonon" which means triangle, and "metria” which means measure. • Trigonon metria = “triangle measurement” • Sine - comes from the Latin word sinus meaning a curve or fold. • Cosine - was originally written "co. sine, " short for COMPLEMENTI SINUS: the sine of the complementary angle. • Tangent - comes from the Latin word tangens which is a straight line which touches a circle at one point.

Triangle Ratios

SOH CAH TOA is mnemonic for: • SOH • CAH • TOA

Trigonometry B a C c b A

EXAMPLE 3 -HOTS THINK…. SHARE

EXAMPLE 3 SOLUTIONS

Angle of Depression 42 ° Angle of Elevation

EXAMPLE 4 – HOTS FINDING A HYPOTENUSE USING AN ANGLE OF DEPRESSION

EXAMPLE 4 SOLUTION

You want to build a skateboard ramp with a length of 14 feet and an angle of elevation of 26°. You need to find the height and length of the base of the ramp. adj. cos 26 o = hyp. opp. sin 26 o = hyp. sin 26 o = x 14 14 sin 26 o = x 6. 1 ft. x cos 26 o = y 14 14 cos 26 o = y 12. 6 ft. y

Using Trigonometry • A 29 c C a B

Try on your own: y opp. tan 32 o = adj. tan 32 o = 11 x x tan 32 o = 11 11 x = tan 32 o 11 x 0. 6249 x 17. 6

Using Trigonometry • A 20 -ft wire supporting a flagpole forms a 35 o angle with the flagpole. To the nearest foot, how high is the flagpole? 20 x

Tallest Mountain in the World? • Mount Everest – 29, 029 feet

Tallest Building in the World? • The Burj Khalifa, Dubai – 2717 feet

The Tallest Waterfall in the World? • Angel, Salto – Venezuela - 3, 212 feet

Clinometer

DEFINITIONS • The angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (the line of sight). • If the object is below the level of the observer, then the angle between the horizontal and the observer's line of sight is called the angle of depression.

How tall is the Empire State Building? Exact Answer: 1, 250 feet tall 76. 5° 300 ft

Since Mount Everest is 29, 029 feet tall, how far was Mr. Sharma from the mountain, if his angle of elevation was 55°? 20, 326. 3 feet

Mr. Sharma is looking up at the waterfall in Angel, Salto – Venezuela, which is 3, 212 feet tall, and has an angle of elevation of 82°. He wants to know the distance that his line of sight makes to the top of the waterfall? 3, 243. 6 feet

Using Trigonometry • R 41 S 47 T