Transcript hypotenuse
Trigonometric Ratios Lesson 12.1 HW: 12.1/1-22 Warm – up Find the missing measures. Write all answers in radical form. x 30° 30 45 60 45 10 z y 60° y x3 3 y3 2 z 5 3 y 5 What is a trigonometric ratio? The relationships between the angles and the sides of a right triangle are expressed in terms of TRIGONOMETRIC RATIOS. We need to do some housekeeping before we can proceed… In trigonometry, the ratio we are talking about is the comparison of the sides of a RIGHT TRIANGLE. Several things MUST BE understood: 1. This is the hypotenuse.. This will ALWAYS be the hypotenuse 2. This is 90°… this makes the right triangle a right triangle…. One more thing… θ and 𝜶 are symbols for unknown angle measures. Their names are ‘Theta’ and ‘Alpha’, from the Greek alphabet. Don’t let it scare you… it’s like ‘x’ except for angle measure… it’s a way for us to keep our variables understandable and organized. The 2 other angles and the 2 other sides adjacent A We will refer to the sides in terms of their proximity to the angle opposite If we look at angle A, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse. Greek letter ‘PHI’ Adjacent side Opposite side opposite If we look at angle B, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse. adjacent B Opposite side Greek Letter ‘Theta’ Adjacent side Remember we won’t use the right angle X Here we go!!!! The Trigonometric Functions we will be looking at SINE COSINE TANGENT The Trigonometric Functions SINE COSINE TANGENT SINE sin Pronounced “sign” COSINE cos Pronounced “co-sign” TANGENT tan Pronounced “tan-gent” The Trigonometric Ratios So, what does this stuff mean?... Adj Leg Cos Hyp Opp Leg Tan Adj Leg adjacent opposite opposite Opp Leg Sin Hyp hypotenuse We need a way to remember all of these ratios… Some Old Hippie Came A Hoppin’ Through Our Old Hippie Apartment Old Hippie Sin Opp Hyp Cos Adj Hyp Tan Opp Adj SOHCAHTOA For any right-angled triangle Definition of Sine Ratio opposite side sin = hypotenuse For any right-angled triangle Definition of Cosine Ratio adjacent side cos = hypotenuse For any right-angled triangle Definition of Tangent Ratio. opposite side tan = adjacent Find: Sin 16° Tan 58° Ex. 1: Finding Trig Ratios Large sin A = cos A = tan A = opposite hypotenuse 8 17 adjacent 15 hypotenuse 17 opposite adjacent 8 Small 4 ≈ 0.4706 8.5 7.5 ≈ 0.8824 8.5 ≈ 0.4706 ≈ 0.8824 4 ≈ 7.5 ≈ 0.5333 15 0.5333 B B 17 8.5 4 8 A A 15 C 7.5 C Trig ratios are often expressed as decimal approximations. Ex. 2: Finding Trig Ratios SohCahToa <S opp sin S = hyp adj cos S = hyp opp tan S = adj 5 0.3846 ≈ 13 12 ≈0.9231 13 5 ≈ 0.4167 12 R opposite 5 13 hypotenuse T 12 adjacent S Finding sin, cos, and tan. (Just writing a ratio or decimal.) Find the sine, the cosine, and the tangent of angle A. Give a fraction and decimal answer (round to 4 places). 9 opp sin A 10.8 .8333 hyp 10.8 9 A adj 6 cos A hyp 10.8 .5556 6 Shrink yourself down and stand where the angle is. opp tan A adj Now, figure out your ratios. 9 6 1.5 Find the sine, the cosine, and the tangent of angle A Give a fraction and decimal answer (round to 4 decimal places). 24.5 8.2 A 23.1 Shrink yourself down and stand where the angle is. 8 .2 opp sin A . 3347 24.5 hyp adj cos A hyp 23.1 24.5 .9429 opp tan A adj 8 .2 23.1 .3550 Now, figure out your ratios. Finding an angle. (Figuring out which ratio to use and getting to use the 2nd button and one of the trig buttons.) Ex. 1: Find . Round to four decimal places. nd 2 17.2 9 17.2 tan 9 tan 17.2 9 ) 62.3789 Shrink yourself down and stand where the angle is. Now, figure out which trig ratio you have and set up the problem. SohCahToa Make sure you are in degree mode (not radians). Ex. 2: Find . Round to three decimal places. 7 23 nd 2 7 cos 23 cos 7 23 ) 72.281 SohCahToa Make sure you are in degree mode (not radians). Finding the angle measure In the figure, find opposite side sin = sin hypotenuse = = 34.85 4 7 4 7 Finding the length of a side In the figure, find y opposite side sin 35 = hypotenuse sin 35 = y 11 11* sin35 = y y= 6.31 y 35° 11 Finding the angle measure In the figure, find cos = cos = = = adjacent Side 3 hypotenuse 3 8 67.98 8 Finding the length of a side In the figure, find x adjacent side cos 42 = hypotenuse cos 42 = x= x= 6 x 6 cos 42 8.07 6 42° x Finding the angle measure In the figure, find 3 tan = tan == opposite side adjacent side 3 5 = 78.69 5 Finding the length of a side In the figure, find z tan 22 = tan 22 = z= z Opposite side adjacent Side 5 z 5 tan 22 z = 12.38 5 22 Conclusion opposite side sin hypotenuse adjacent side cos hypotenuse opposite side tan adjacent side Make Sure that the triangle is right-angled SohCahToa Solving a Problem with the Tangent Ratio SohCahToa h=? 60º 53 ft We know the angle and the side adjacent to 60º. We want to know the opposite side. Use the tangent ratio: opp h tan 60 adj 53 h tan 60 53 h 53 tan 60 h 53 3 92 ft Note: • The value of a trigonometric ratio depends only on the measure of the acute angle, not on the particular right triangle that is used to compute the value. Ex. 4: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 30 30 sin 30= opposite hypotenuse adjacent cos 30= hypotenuse opposite tan 30= adjacent 1 2 = 0.5 √3 ≈ 0.8660 2 1 = √3 √3 3 ≈ 0.5774 Begin by sketching a 30-60-90 triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. From Theorem 9.9, on page 551, it follows that the length of the longer leg is √3 and the length of the hypotenuse is 2. 2 1 30 √3 Ex: 5 Using a Calculator • You can use a calculator to approximate the sine, cosine, and the tangent of 74. Make sure that your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators. Sample keystrokes Sample keystroke sequences 74 sin or sin 74 Sample calculator display Rounded Approximation 0.961262695 0.9613 0.275637355 0.2756 3.487414444 3.4874 ENTER 74 or COS COS 74 ENTER 74 TAN or TAN 74 ENTER Notes: • If you look back at Examples 1-5, you will notice that the sine or the cosine of an acute triangles is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse. The length of a leg or a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one.