4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles

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1 4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. The sides a and b are referred to as the legs of the right triangle. Upper case A, B and C are its angles with C being the right angle. A+ B+ C 180 (True for all triangles) A and B are complementary angles ( A+ B 90 ) Pythagorean Theorem: 2 2 a + b 2 c... and all of its rearrangements c 2 2 a + b (can t simplify to c a+ b... TRAGIC mistake!) a 2 2 c b b 2 2 c a Handy Pythagorean Triples: just keep in mind when using Pythagorean triples that the hypotenuse MUST be the longest side of the right triangle.

2 Similar Right Triangles Two right triangles are similar if their matching sides are in the same proportion. The measure of each angle (using a protractor): α β θ alpha beta theta The trigonometric functions are based on this premise: the ratio of any two sides of similar right triangles can be paired with the angle measures they share. These ratios are a function of the angle... NOT a function of the triangle! There are 6 such ratios you can create using the sides of a right triangle.

3 THE SIX TRIGONOMETRIC RATIOS ADJ* adjacent leg to angle θ OPP* opposite leg to angle θ HYP hypotenuse (* The role of ADJ and OPP depend on which of the two acute angles is being used) The main three: sin( θ ) OPP HYP Pronounced " Sine theta" cos( θ ) ADJ HYP " Cosine theta" tan( θ ) OPP ADJ " Tangent theta" csc( θ )... and their reciprocals: HYP OPP " Cosecant theta" HYP sec( θ ) cot( θ ) ADJ " Secant theta" ADJ OPP "Cotangent theta" A handy way to remember the main three ratios is to use SOH CAH TOA. S O H stands for Sine O / H C A H stands for Cosine A / H T O A stands for Tangent O / A The reciprocals need to be committed to memory. For sine and cosine s reciprocals always remember to pair an S with a C : Sine s reciprocal is Cosecant Cosine s reciprocal is Secant It should be easy to remember that tangent and cotangent are reciprocals.

4 ex) Evaluate all 6 trigonometric ratios for the angle θ shown in the diagram. (You ll need to determine the missing value first.) sin( θ) csc( θ) cos( θ) sec( θ) tan( θ) cot( θ) ex) Evaluate all 6 trigonometric ratios for the angle θ shown in the diagram. ALWAYS MAKE SURE YOU RATIONALIZE DENOMINATORS! sin( θ) csc( θ) cos( θ) sec( θ) tan( θ) cot( θ)

5 What does the CO stand for? In the triangle shown here, remember that angles α and β are complementary. Evaluate sin( α) and cos( β) Evaluate tan( α) and cot( β) Evaluate sec( α) and csc( β) For any pair of complementary angles, a trig ratio and its co function counterpart have the same value. ex) The value of sin(28 ) is equivalent to the cos( ) The value of cot(61 ) is equivalent to the tan( ) Solving Right Triangle Problems (with a calculator) The main three trig ratios are built into your calculator. MAKE SURE YOUR CALCULATOR IS IN THE CORRECT ANGLE MODE! ex) Evaluate cos(28 ) and cos(28)

6 To solve a right triangle means to provide all the missing measurements. You ll need to take the given information and relate a trigonometric value of an angle to a ratio of a known and unknown side. ex) Solve the triangle. Round the side measurements to 2 decimal places. ex) A service ramp makes an angle of elevation measuring The ramp needs to accommodate a vertical rise of 1 meter. How long should the ramp be? Round to the nearest tenth of a meter.

7 ex) A ship leaves harbor at a bearing of S 75 E at a speed of 50 mph. After 2 hours without changing course, how far south of the harbor has the boat traveled? Round to the nearest tenth of a mile. What are SIN 1, COS 1 and TAN 1? The second functions above SIN, COS and TAN buttons are for determining angles based on a known sine, cosine or tangent ratio. ex) If cos( θ ) 0.32 ex) Determine the angle shown here. What is the value of θ in degrees? What is the value of θ in radians?

8 Special Angle Right Triangles... YOU NEED TO KNOW THESE! For 30 or π 6 For π 45 or 4 For π 60 or 3 These triangles can take any size BUT the sides are always proportional to these values. For example: for a triangle, the ratio of the short leg to the hypotenuse will ALWAYS be 1:2

9 ex) Evaluate the following WITHOUT using your calculator. sin(30 ) tan(45 ) sec(π/6) sin(π/4) sec(60 ) tan(π/6) csc(45 ) cos(π/3) sin(π/4) cos(30 ) cot(60 ) csc(π/3)