#### Transcript Trigonometric Functions

Trigonometric Ratios MM2G2. Students will define and apply sine, cosine, and tangent ratios to right triangles. MM2G2a: Discover the relationship of the trigonometric ratios for similar triangles. Trigonometric Ratios MM2G2b: Explain the relationship between the trigonometric ratios of complementary angles. MM2G2c: Solve application problems using the trigonometric ratios. The following slides have been come from the following sources: www.mccd.edu/faculty/bruleym/.../trigo nometric%20ratios http://ux.brookdalecc.edu/fac/cos/lschm elz/Math%20151/ www.scarsdaleschools.k12.ny.us /202120915213753693/lib/…/trig.ppt Emily Freeman McEachern High School Warm Up Put 4 30-60-90 triangles with the following sides listed and have students determine the missing lengths. 30 S 5 2 7√3 √2 90 H 10 4 14√3 2√2 60 L 5√3 2√3 21 √6 Trigonometric Ratios Talk about adjacent and opposite sides: have the kids line up on the wall and pass something from one to another adjacent and opposite in the room. Make a string triangle and talk about adjacent and opposite some more Trigonometric Ratios Determine the ratios of all the triangles on the board and realize there are only 3 (6?) different ratios. Talk about what it means for shapes to be similar. Make more similar right triangles on dot paper, measure the sides, and calculate the ratios. Trigonometric Ratios Try to have the students measure the angles of the triangles they made on dot paper. Do a Geosketch of all possible triangles and show the ratios are the same for similar triangles Finally: name the ratios Warm Up Pick up a sheet of dot paper, a ruler, and protractor from the front desk. Draw two triangles, one with sides 3 & 4, and the other with sides 12 & 5 Calculate the hypotenuse Calculate sine, cosine, and tangent for the acute angles. Measure the acute angles to the nearest degree. Show how to find sine, cosine, & tangent of angles in the calculator Yesterday We learned the sine, cosine, and tangent of the same angle of similar triangles are the same Another way of saying this is: The sine, cosine, tangent of congruent angles are the same Trigonometric Ratios in Right Triangles M. Bruley Trigonometric Ratios are based on the Concept of Similar Triangles! All 45º- 45º- 90º Triangles are Similar! 2 45 º 1 1 45 º 1 2 1 2 1 2 2 45 º 2 2 All 30º- 60º- 90º Triangles are Similar! 2 30º 3 30º 60º 4 2 3 1 60º 1 30º 60º ½ 2 3 2 All 30º- 60º- 90º Triangles are Similar! 2 60º 10 1 30º 30º 3 5 3 1 60º 30º 3 2 1 2 60º 5 In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuse We’ll label them a, b, and c and the angles and . Trigonometric functions are defined by taking the ratios of sides of a right triangle. adjacent c First let’s look at the three basic functions. leg b SINE leg a COSINE TANGENT They are abbreviated using their first 3 letters opposite a opposite a sin tan hypotenuse c adjacent b adjacent b cos hypotenuse c The Trigonometric Functions SINE COSINE TANGENT SINE Prounounced “sign” COSINE Prounounced “co-sign” TANGENT Prounounced “tan-gent” Greek Letter q Pronounced “theta” Represents an unknown angle Greek Letter α Pronounced “alpha” Represents an unknown angle Greek Letter β Pronounced “Beta” Represents an unknown angle Opp Sin Hyp hypotenuse hypotenuse Adj Cos Hyp Opp Tan Adj q adjacent opposite We could ask for the trig functions of the angle by using the definitions. You MUST get them memorized. Here is a mnemonic to help you. c The sacred Jedi word: b SOHCAHTOA adjacent a opposite b sin hypotenuse c adjacent a cos hypotenuse c opposite b tan adjacent a It is important to note WHICH angle you are talking about when you find the value of the trig function. c 5 4b Let's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so a b c 2 2 2 adjacent a3 sin = o 3 h 5 Let's choose: tan = o 4 a 3 32 42 52 Use a mnemonic and figure out which sides of the triangle you need for tangent. sine. You need to pay attention to which angle you want the trig function of so you know which side is opposite that angle and which side is adjacent to it. The hypotenuse will always be the longest side and will always be opposite the right angle. Oh, I'm acute! This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle. 5 4 3 So am I! We need a way to remember all of these ratios… What is SohCahToa? Is it in a tree, is it in a car, is it in the sky or is it from the deep blue sea ? This is an example of a sentence using the word SohCahToa. I kicked a chair in the middle of the night and my first thought was I need to SohCahToa. An example of an acronym for SohCahToa. Seven old horses Crawled a hill To our attic.. Some Old Hippie Came A Hoppin’ Through Our Old Hippie Apartment SOHCAHTOA Old Hippie Sin Opp Hyp Cos Adj Hyp Tan Opp Adj Other ways to remember SOH CAH TOA 1. Some Of Her Children Are Having Trouble Over Algebra. 2. Some Out-Houses Can Actually Have Totally Odorless Aromas. 3. She Offered Her Cat A Heaping Teaspoon Of Acid. 4. Soaring Over Haiti, Courageous Amelia Hit The Ocean And ... 5. Tom's Old Aunt Sat On Her Chair And Hollered. -- (from Ann Azevedo) Other ways to remember SOH CAH TOA 1. Stamp Out Homework Carefully, As Having Teachers Omit Assignments. 2. Some Old Horse Caught Another Horse Taking Oats Away. 3. Some Old Hippie Caught Another Hippie Tripping On Apples. 4. School! Oh How Can Anyone Have Trouble Over Academics. Trigonometry Ratios opposite Tangent A = adjacent opposite Sine A = hypotenuse adjacent Cosine A = hypotenuse Soh Cah Toa A 14º 24º 60.5º 46º 82º The Tangent of an angle is the ratio of the opposite side of a triangle to its adjacent side. hypotenuse 1.9 cm opposite adjacent 14º 7.7 cm 1.9 7.7 0.25 Tangent 14º 0.25 Tangent A = opposite adjacent 3.2 cm 7.2 cm 3.2 7.2 0.45 24º Tangent 24º 0.45 Tangent A = opposite adjacent 5.5 cm 46º 5.5 5.3 5.3 cm 1.04 Tangent 46º 1.04 Tangent A = 6.7 cm 6.7 3.8 opposite adjacent 1.76 Tangent 60.5º 1.76 60.5º 3.8 cm Tangent A = opposite adjacent As an acute angle of a triangle approaches 90º, its tangent becomes infinitely large very large Tan 89.9º = 573 Tan 89.99º = 5,730 etc. very small Since the sine and cosine functions always have the hypotenuse as the denominator, and since the hypotenuse is the longest side, these two functions will always be less than 1. opposite Sine A = hypotenuse adjacent Cosine A = hypotenuse Sine 89º = .9998 A Sine 89.9º = .999998 Sin α = opposite hypotenuse 7.9 cm 3.2 cm 24º 3 .2 7 .9 0.41 Sin 24º 0.41 adjacent Cosine β = hypotenuse 7.9 cm 46º 5 .5 7 .9 5.5 cm 0.70 Cos 46º 0.70 A plane takes off from an airport an an angle of 18º and a speed of 240 mph. Continuing at this speed and angle, what is the altitude of the plane after 1 minute? After 60 sec., at 240 mph, the plane has traveled 4 miles x 4 18º SohCahToa Soh Sine A = opposite hypotenuse x opposite x Sine 18 = 4 x 0.3090 = 4 1 x = 1.236 miles or 4 6,526 feet hypotenuse 18º An explorer is standing 14.3 miles from the base of Mount Everest below its highest peak. His angle of elevation to the peak is 21º. What is the number of feet from the base of Mount Everest to its peak? x x Tan 21 = 0.3839 = 14.3 1 14.3 x x = 5.49 miles = 29,000 feet 14.3 21º A swimmer sees the top of a lighthouse on the edge of shore at an 18º angle. The lighthouse is 150 feet high. What is the number of feet from the swimmer to the shore? 150 150 Tan 18 = x 0.3249 =150 x 1 x 0.3249x = 150 0.3249 0.3249 X = 461.7 ft 18º A dragon sits atop a castle 60 feet high. An archer stands 120 feet from the point on the ground directly below the dragon. At what angle does the archer need to aim his arrow to slay the dragon? 60 Tan x = 120 Tan x = 0.5 Tan-1(0.5) = 26.6º 60 x 120 Solving a Problem with the Tangent Ratio We know the angle and the side adjacent to 60º. We want to know the opposite side. Use the tangent ratio: h=? 2 3 60º 53 ft 1 tan 60 opp h adj 53 3 h Why? 1 53 h 53 3 92 ft Ex. A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree? tan 71.5° ? 50 71.5 ° Opp Hyp y 50 tan 71.5° y = 50 (tan 71.5°) y = 50 (2.98868) y 149.4 ft Ex. 5 A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge? cos 60° x (cos 60°) = 200 200 60° x x X = 400 yards Trigonometric Functions on a Rectangular Coordinate System y Pick a point on the terminal ray and drop a perpendicular to the x-axis. r y q x x The adjacent side is x The opposite side is y The hypotenuse is labeled r This is called a REFERENCE TRIANGLE. y r x cos q r y tan q x sin q r y r sec q x x cot q y csc q Trigonometric Ratios may be found by: Using ratios of special triangles 2 45 º 1 1 1 2 1 cos 45 2 tan 45 1 sin 45 For angles other than 45º, 30º, 60º you will need to use a calculator. (Set it in Degree Mode for now.)