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ANCIENT GREEKS USED TRIGONOMETRY TO MEASURE THE DISTANCE TO THE STARS IN 140 B.C. HIPPARCHUS BEGAN TO USE AND WRITE TRIGONOMETRY TRIGONOMETRY GREEK WORD MEANING: TRIANGLE MEASURE Trigonometry is only used for RIGHT TRIANGLES 90 RIGHT TRIANGLES MUST HAVE A 90 DEGREE ANGLE Definitions • Right Triangle – a triangle with one right angle and two acute angles. • Hypotenuse – the longest side in a right triangle. The hypotenuse is always across from the right angle. • Legs – the two other sides in a right triangle that form an “L” and create the right angle. • Opposite Side – The side across from the angle we are talking about. • Adjacent Side – The side next to the angle we are talking about. LEG OPPOSITE TO B B LEG ADJACENT TO ANGLE B Now that we know the pieces of the right triangle…. Let’s learn about the trigonometric functions. The Trigonometric Functions we will be looking at SINE COSINE TANGENT The Trigonometric Functions are shortened on the calculator to the colored part of the word. SINE COSINE TANGENT SINE Prounounced “sign” COSINE Prounounced “co-sign” TANGENT Prounounced “tan-gent” Greek Letter q Prounounced “theta” Represents an unknown angle LEG OPPOSITE TO θ LEG ADJACENT TO θ θ SINE OF θ =LENGTH OF LEG OPPOSITE θ LENGTH OF HYPOTENUSE COSINE OF θ = LENGTH OF LEG ADJACENT θ LENGTH OF HYPOTENUSE TANGENT OF θ = LENGTH OF LEG OPPOSITE θ LENGTH OF LEG ADJACENT TO θ We need a way to remember all of these ratios… S O H C A H T O A SOHCAHTOA Will help us remember these equations. Sine Opposite Hypotenuse Cosine Adjcent Hypotenuse Tangent Opposite Adjcent Once Upon there was a mighty Chief SohCahToa. He fought hard to protect and fix the damaged tee-pee’s, but they were missing one side…OH NO! ? He realized how he could fix his tee-pee’s by using trigonometry. SohCahToa OPP SIN= HYP OPP TAN= ADJ ADJ COS= HYP ? It is all in my name. If I can label the sides of a triangle I can use that to fix the missing sides of these tee-pees. Soh OPPOSITE THE ANGLE OPP SIN= HYP X ADJ COS= HYP X ADJACENT TO THE ANGLE Cah OPPOSITE THE ANGLE OPP TAN= ADJ X ADJACENT TO THE ANGLE Toa They’re Fixed !! Thank goodness for Chief SohCahToa. Opp sin q Hyp hypotenuse opposite opposite Adj cos q Hyp q Opp tan q Adj adjacent Finding sin, cos, and tan Here are some notes and examples: When I have a right triangle … To find the missing side I am going to create a proportion. Ex. 1. 2. 3. 4. 5. sin q o 1 h Mark the angle that is given. Label the sides adjacent, opposite and hypotenuse. Ask myself…what side am I trying to find? What side am I given? Which trigonometric function uses those two sides? Create your proportion. Let’s Do some examples: • Copy these onto your blank space under examples. Create your proportion! Opp Sin q Hyp 8 10 4 5 Reduce . 10 Reduce . 3 6 10 5 Create your proportion! Opposite Create your proportion! Adj Cosq Hyp SOHCAHTOA Reduce . Opp 8 4 Tanq Adj 6 3 q Mark the Angle Adjacent 6 Label the sides… 8 Find the sine, the cosine, and the tangent of angle A. Give a fraction and decimal answer (round to 4 places). Create your proportion! Opposite 9 9 opp sin A hypo 10.8 10.8 .8333 Create your proportion! A Adjacent 6 Mark the Angle Label the sides… adj 6 cos A hypo 10.8 .5555 Create your proportion! opp tan A adj 9 6 1.5 Find the sine, the cosine, and the tangent of angle A Adjacent 23.1 Opposite A 24.5 8.2 Create your proportion! B Mark the Angle Label the sides… Give a fraction and decimal answer (round to 4 decimal places). opp 8.2 sin A .3347 24 . 5 hyp Create your proportion! adj cos A hyp 23.1 24.5 .9429 Create your proportion! opp tan A adj 8 .2 23.1 .3550 You are done with this part. • Close out of this screen. • Put your laptop away (make sure it is plugged in) • Come back to your seat and try to complete the rest of the packet using your notes and examples.