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.
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.
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.
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)
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)
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 10.75. 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.
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?
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 30 60 90 triangle, the ratio of the short leg to the hypotenuse will ALWAYS be 1:2
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)