Right Triangle Trigonometry:

Relating to the Real World Before any spacecraft ever traveled to another planet, astronomers had figured out the distance from each planet to the sun. They accomplished this feat by using trigonometry the mathematics of triangle measurement. You will learn how to use trigonometry to measure distances that you could never otherwise measure.

Measure for Measure What do the Trigonometric Ratios tell you about the parts of a triangle? What are the conditions for triangles to be similar? How can you remember the Trigonometric ratios?

Label These Two Triangles Angle A Angle A

The Tangent Ratio TANGENT OF A = leg opposite A leg adjacent to A The word Trigonometry comes from the Greek words meaning “triangle measurement.” A ratio of the lengths of sides of a right triangle. This ratio is called the TANGENT. TANGENT OF A = leg opposite A leg adjacent to A Slope of a line = Rise Run

Using the Tangent Ratio Tangent of < A = Leg OPPOSITE TO < A Leg ADAJACENT TO <A THIS EQUATION CAN BE ABBREVIATED AS: TAN A = OPPOSITE = BC ADJACENT CA B Leg Opposite to < A, BC C A Leg Adjacent to < A, CA

Example Write the Tangent Ratios for < U and < T. Tan U = opposite = TV = 3 adjacent UV 4 T 5 Tan T = opposite = UV = 4 adjacent TV 3 3 V 4 U

Try This Write the Tangent ratios for < K and < J How is Tan K related to the Tan J ? J 3 K L 7

Try This: Find the Tangent of < A to the nearest tenth 1. 2. Hint: Find the Ratio First! A 4 8 5 A 4

Using the Sine Ratio Sine of < A = Leg OPPOSITE TO < A Hypotenuse THIS EQUATION CAN BE ABBREVIATED AS: Sin A = OPPOSITE = BC HYPOTENUSE AB Sin B = OPPOSITE = CA B Leg Opposite to < A, BC Hypotenuse, AB C A Leg Opposite to < B, CA

Example Write the Sine Ratios for < U and < T. Sin U = opposite = TV = 3 hypotenuse TU 5 T 5 Sin T = opposite = UV = 4 hypotenuse TU 5 3 V 4 U

Try This Write the Sine ratios for < K and < J J 10 6 K L 8

Using the Cosine Ratio Cosine of < A = Leg Adjacent TO < A Hypotenuse THIS EQUATION CAN BE ABBREVIATED AS: Cos A = ADJACENT = CA HYPOTENUSE AB Cos B = ADJACENT = BC B Leg Adjacent to < B, BC Hypotenuse, AB C A Leg Adjacent to < A, CA

Example Write the Cosine Ratios for < U and < T. Cos U = adjacent = UV = 4 hypotenuse TU 5 T 5 Cos T = adjacent = TV = 3 hypotenuse TU 5 3 V 4 U

Try This Write the Cosine ratios for < K and < J J 10 6 K L 8

Some Old Hippie Came A Hoppin’ Through Our Old Hippie Apartment

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

Ex: 1. Figure out which ratio to use. Find x Ex: 1 Figure out which ratio to use. Find x. Round to the nearest tenth. 20 m x Shrink yourself down and stand where the angle is. tan 20 55 ) Now, figure out which trig ratio you have and set up the problem.

Ex: 2 Find the missing side. Round to the nearest tenth. 80 ft x tan 80  ( 72 ) ) = Shrink yourself down and stand where the angle is. Now, figure out which trig ratio you have and set up the problem.

Ex: 3 Find the missing side. Round to the nearest tenth. Shrink yourself down and stand where the angle is. Now, figure out which trig ratio you have and set up the problem.