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153 Section 10.6 – Right Triangle Trigonometry Objective #1: Understanding Adjacent, Hypotenuse, and Opposite sides of an acute angle in a right triangle. In a right triangle, the hypotenuse is always the longest side; it is the side opposite of (or "across from") the right angle. The hypotenuse of the following triangles is highlighted in gray. The adjacent side of an acute angle is the side of the triangle that forms the acute angle with the hypotenuse. The adjacent side is the side of the angle that is "adjacent" to or next to the hypotenuse. The adjacent side depends on which of the acute angles one is using. The adjacent side of the following angles (labeled A) is highlighted in gray. A A A A The opposite side of an acute angle is the side of the triangle that is not a side of the acute angle. It is "across from" or opposite of the acute angle. The opposite side depends on which of the acute angles one is using. The opposite side of the following angles (labeled A) is highlighted in gray. A A A A 154 For each acute angle in the following right triangles, identify the adjacent, hypotenuse, and opposite side: R Ex. 1 B Ex. 2 Q A C X Solution: For ∠R, the adjacent side is QR, the opposite side is QX, and the hypotenuse is RX. For ∠X the adjacent side is QX, the opposite side is QR, and the hypotenuse is RX. Solution: For ∠A, the adjacent side is AC, the opposite side is BC, and the hypotenuse is AB. For ∠B, the adjacent side is BC, the opposite side is AC, and the hypotenuse is AB. Objective #2: Understanding Trigonometric Ratios or Functions. Recall that two triangles are similar if they have the same shape, but are not necessarily the same size. In similar triangles, the corresponding angles are equal. Thus, if ∆ABC ∼ ∆HET, then m∠A = m∠H, m∠B = m∠E, and m∠C = m∠T. The corresponding sides are not equal. However, the ratios of corresponding sides are equal since one triangle is in proportion to the other triangle. Thus, if ∆ABC ∼ ∆HET, then B E AB BC AC = = HE ET HT Now, let's consider the proportion AB BC = . If we cross multiply, HE ET A C we get (AB)•(ET) = (HE)•(BC). Now, H divide both sides by (BC)•(ET) and simplifying, we get: (AB)•(ET) = (HE)•(BC) AB = HE (BC)•(ET) (BC)•(ET) BC ET T 155 Now consider angles ∠C and ∠T. If ∠A and ∠H are right angles, then the proportion AB BC = HE ET B E implies that the ratio of the opposite side to the hypotenuse is the same for any right triangle if the corresponding acute A C angles have the same measure. H T We will define this ratio as the Sine Ratio or Sine Function. The abbreviation for the sine function is "sin", but it is pronounced as "sign". Sine Ratio or Function Given an acute angle A in a right triangle, then the sine ratio of A, is the ratio of the opposite side of A to the hypotenuse. oppositeside opp sin A = = hypotenuse hyp A opp adj hyp For each acute angle in the following right triangles, find the sine of the angle: 16.1 in Ex. 3 24.1 cm Ex. 4 55˚ 50˚ 42.0 cm 19.2 in 25.0 in 35˚ 34.4 cm Solution: opp 24.1 sin 35˚ = = Solution: sin 40˚ = = 0.5738… ≈ 0.574 opp 34.4 sin 55˚ = = = 0.644 sin 50˚ = = 0.8190… ≈ 0.819 = 0.768 hyp hyp 42.0 42.0 40˚ opp hyp = 16.1 25.0 opp hyp = 19.2 25.0 Since the opposite is never greater than the hypotenuse, then the value of the sine function cannot exceed 1. If the opposite side is equal to the hypotenuse, then the measure of the angle is 90˚ and the triangle degenerates into a vertical line. This means that sin 90˚ = 1. Similarly, if the opposite side has zero length meaning the sine ratio is 0, then the measure of the angle is 0˚ and the triangle degenerates into a horizontal line. Hence, sin 0˚ = 0. 156 We have seen that if the corresponding acute angles of rights triangles have the same measure, the ratio of the opposite side to the hypotenuse is fixed. We can make the same argument that the ratio of the adjacent side to the hypotenuse is also fixed. We will define this ratio as the Cosine Ratio or Cosine Function. The abbreviation for the cosine function is "cos", but it is pronounced as "co-sign". Cosine Ratio or Function Given an acute angle A in a right triangle, then the cosine ratio of A, is the ratio of the adjacent side of A to the hypotenuse. cos A = adjacentside hypotenuse = hyp A adj hyp opp adj For each acute angle in the following right triangles, find the cosine of the angle: 16.1 in Ex. 5 24.1 cm Ex. 6 55˚ 50˚ 42.0 cm 19.2 in 25.0 in 35˚ 34.4 cm Solution: adj 34.4 cos 35˚ = = Solution: cos 40˚ = = 0.8190… ≈ 0.819 = 0.768 hyp cos 55˚ = adj hyp 42.0 = 24.1 42.0 = 0.5738… ≈ 0.574 cos 50˚ = 40˚ adj hyp = 19.2 25.0 adj hyp = 16.1 25.0 = 0.644 Since the adjacent is never greater than the hypotenuse, then the value of the cosine function cannot exceed 1. If the adjacent side is equal to the hypotenuse, then the measure of the angle is 0˚ and the triangle degenerates into a horizontal line. This means that cos 0˚ = 1. Similarly, if the adjacent side has zero length meaning the cosine ratio is 0, then the measure of the angle is 90˚ and the triangle degenerates into a vertical line. Hence, cos 90˚ = 0. We have seen that if the corresponding acute angles of rights triangles have the same measure, the ratio of the opposite side to the hypotenuse is 157 fixed and the ratio of the adjacent side to the hypotenuse is fixed. We can make the same argument that the ratio of the opposite side to the adjacent side is also fixed. We will define this ratio as the Tangent Ratio or Tangent Function. The abbreviation for the tangent function is "tan", but it is pronounced as "tangent". Tangent Ratio or Function Given an acute angle A in a right triangle, then the tangent ratio of A, is the ratio of the opposite side of A to the adjacent side of A. oppositeside opp tan A = = adjacentside hyp A opp adj adj For each acute angle in the following right triangles, find the tangent of the angle: 16.1 in Ex. 7 24.1 cm Ex. 8 55˚ 42.0 cm 34.4 cm Solution: opp tan 35˚ = = adj 24.1 34.4 = 0.7005… ≈ 0.701 opp adj = = 1.427… ≈ 1.43 19.2 in 25.0 in 35˚ tan 55˚ = 50˚ 34.4 24.1 Solution: tan 40˚ = opp adj 40˚ = 16.1 19.2 ≈ 0.8385… = 0.839 tan 50˚ = opp adj = 19.2 16.1 = 1.192… ≈ 1.19 When the opposite side has zero length, then the measure of the angle is zero. The tangent ratio is the ratio of 0 to the adjacent side, which is equal to zero. Hence, tan 0˚ = 0. Similarly, if the adjacent side has zero length, then the measure of the angle is 90˚. The tangent ratio is the ratio of the opposite side to 0, which is undefined. Thus, tan 90˚ is undefined. Unlike the sine and cosine function, the values of the tangent function can exceed one. There are some memories aids that you can use to remember the trigonometric ratios. If you remember the order sine, cosine, and tangent, then you might find one of these phrases helpful: 158 Trigonometric Ratio Ratio of sides Rated G Phrase Rated PG-13 Phrase opp hyp adj hyp opp adj oscar had a heap of apples oh h∗ll another hour of algebra sin A = cos A = tan A = If you want to include the trigonometric functions themselves, you can use the following phrases: s-o-h-c-a-h-t-o-a s i n e o p p o s i t e h y p o t e n u s e c o s i n e a d j a c e n t h y p o t e n u s e t a n g e n t o p p o s i t e a d j a c e n t sinful oscar had consumed a heaping taste of apples sinful i n e € € € oscar p p o s i t e had y p o t e n u s e consumed o s i n e a d j a c e n t heaping y p o t e n u s e taste a n g e n t of p p o s i t e apples. d j a c e n t 159 Objective #3: Evaluating trigonometric ratios on a scientific calculator. On a scientific calculator, SIN key is for the sine ratio, COS key is for the cosine ratio, and TAN key is for the tangent ratio. To evaluate a trigonometric ratio for a specific angle, you first want to be in the correct mode. If the angle is in degree, then the calculator needs to be in degree mode. Hit the appropriate trigonometric ratio, type the angle and hit enter. Evaluate the following: Ex. 9a Ex. 9c cos 26˚ 2 sin 16 ˚ Ex. 9b tan 78.3˚ 3 Solution: a) We put our calculator in degree mode, hit the COS key, type 26 and hit enter: cos 26˚ = 0.8987… ≈ 0.899 b) We put our calculator in degree mode, hit the TAN key, type 78.3 and hit enter: tan 78.3˚ = 4.828… ≈ 4.83 c) We put our calculator in degree mode, hit the SIN key, type 2 2 16 and hit enter: sin 16 ˚ = 0.2868… ≈ 0.287 3 Objective #4: 3 Finding Angles using Inverse Trigonometric Ratio. If we know the ratio of two sides of a right triangle, we can use inverse trigonometric ratios or functions to find the angle. The inverse trigonometric function produces an angle for the answer. Inverse Trigonometric Functions: Given a right triangle, we can find the acute angle A using any of the following: 1) Inverse sine (arcsine) opp A = sin – 1( ) hyp A opp adj hyp 2) Inverse cosine (arccosine) adj A = cos – 1( ) hyp 3) Inverse tangent (arctangent) A = tan – 1( opp adj ) On a scientific calculator, these functions are labeled SIN – 1, COS – 1 and TAN – 1. They are usually directly above the SIN, COS, and TAN keys. To 160 find the angle on the calculator, first set your calculator in degree mode. Second, hit the 2nd (or INV) and then the appropriate trigonometric ratio key followed by the ratio of the two sides you are given and hit enter. Solve the following. Round to the nearest tenth: Ex. 10a sin x = 0.833 Ex. 10c tan x = 1.82 Ex. 10b cos x = 0.528 Solution: a) Since sin x = 0.833, then x = sin – 1(0.833). Hit 2nd SIN, type 0.833 and hit enter: sin – 1(0.833) ≈ 56.4˚ b) Since cos x = 0.528, then x = cos – 1(0.528). Hit 2nd COS, type 0.528 and hit enter: cos – 1(0.528) ≈ 58.1˚ c) Since tan x = 1.82, then x = tan – 1(1.82). Hit 2nd TAN, type 1.82 and hit enter: tan – 1(1.82) ≈ 61.2˚ Objective #5: Finding the missing sides and angles of a right triangle. Given either two sides or a side and one of the acute angles of a right triangle, we can use our trigonometric ratios and the Pythagorean Theorem to solve for the missing sides and angles of a right triangle. Finding the missing sides and angles of the following. Round all sides and angles to the nearest tenth: E Ex. 11 Ex. 12 B 18 ft 71.5˚ 23˚ C A Solution: Since ∠A and ∠B are complimentary, then m∠B = 90˚ – 23˚ = 67˚. We have the hypotenuse and the acute angle A. We can use the sine of 23˚ to find BC and the cosine of 23˚ to find AC. Solution: F 13 in G Since ∠E and ∠G are complimentary, then m∠E = 90˚ – 71.5˚ = 18.5˚ = 18˚. We have the adjacent side and and the acute angle G. We can use the tangent of 71.5˚ to find the EF and the cosine of 71.5˚ to find EG. € 161 sin 23˚ = opp hyp = BC 18 opp adj tan 71.5˚ = EF 13 = To solve, multiply by 18: BC = 18•sin 23˚ = 18•0.3907… = 7.03… € ≈€7.0 ft adj AC cos 23˚ = = To solve, multiply by 13: EF = 13•tan 71.5˚ = 13•2.98… = 38.85… € in € ≈ 38.9 adj 13 cos 71.5˚ = = To solve for BC, multiply by 18: AC = 18•cos 23˚ € =€18•0.9205… = 16.56… ≈ 16.6 ft Thus, BC ≈ 7.0 ft. AC ≈ 16.6 ft and m∠B = 67˚. To solve for EG, multiply by EG and then divide by cos 71.5˚: EG•cos 71.5˚ = 13 EG = 13/cos 71.5˚ =13/0.317… = 40.97… ≈ 41.0 in Hence, EF ≈ 38.9 in, EG ≈ 41.0 in, and m∠E = 18.5˚. hyp Ex. 13 C hyp 18 7 cm B Ex. 14 F 10.1 m 5 cm E A Solution: We have the two legs of a right triangle. We can use the inverse tangent function to find ∠B: . tan B = EG opp adj = 5 7 Thus, B = tan – 1( Solution: We have the leg and the hypotenuse of a right triangle. we can use the inverse cosine to to find ∠G. cos G = 5 7 ) ≈ 35.5˚ Since ∠A and ∠B are € €complimentary, then m∠A = 90˚ – 35.5˚ = 54.5˚ € To find the length AB, we can use the Pythagorean Theorem: (AB)2 = (7)2 + (5)2 (AB)2 = 49 + 25 (AB)2 = 49 + 25 G 15 m adj hyp = 10.1 15 Hence, G = cos – 1( 10.1 ) 15 ≈ 47.7˚ Since ∠E and ∠G are € €complimentary, then m∠E = 90˚ – 47.7˚ = 42.3˚ € To find the length EF, we can use the Pythagorean Theorem: (10.1)2 + (EF)2 = (15)2 102.01 + (EF)2 = 225 subtract 102.01 from both sides 162 (AB)2 = 74 AB = 74 = 8.60… ≈ 8.6 cm Thus, m∠B ≈ 35.5˚, m∠A ≈ 54.5˚, and AB ≈ 8.6 cm. € Objective #6: (EF)2 = 122.99 EF = 122.99 =11.09… ≈ 11.1 m Hence, m∠G ≈ 47.7˚, m∠E = 42.3˚, and EF ≈ 11.1 m. € Solving applications involving right triangles. The key to solving applications involving right triangles is drawing a reasonable diagram to represent the situation. Next, we need determine what information we are given and what information we need to find. Finally, we then use the appropriate trigonometric function to solve the problem. Solve the following: Ex. 15 A surveyor measures the angle between her instruments and the top of a tree and finds the angle of elevation to be 65.7˚. If her instruments are 4 ft high off of the ground and are 25 feet away from the center of the base of the tree and the ground is leveled, how high is the tree? Solution: We begin by drawing the surveyor's instruments and the tree. Since her instruments are 4 ft above the ground, we can draw a horizontal line from her her instruments to four feet above the center of the base of the tree. Drawing x a line from the top of the tree to her instruments and a vertical line down 65.7˚ from the top of the tree four feet above 25 feet the center of the base of the tree, we get a right triangle. The adjacent side is 25 feet and we are looking for the opposite of the triangle. Once we get the opposite side, we will need to add 4 ft to the answer to find the height of the tree. The trigonometric ratio that uses the opposite and adjacent sides is the tangent function. Hence, opp x tan 65.7˚ = = (multiply by 25 to solve for x) adj 25 x = 25•tan 65.7˚ = 25•2.214… = 55.36… ft Now, add 4 ft to find the height of the tree: 55.36… + 4 = 59.36 ≈ 59.4 ft So, the tree is approximately 59.4 ft tall. 163 Ex. 16 A helicopter flying directly above an ocean liner at a height of 1400 ft, locates another boat at a 23˚ angle of depression. How far apart are the two ships? Round to the nearest hundred feet. Solution: The angle of depression is the angle between a horizontal line and the line of observation. In other words, it is the angle one has to look down to see the observed object, which is a ship in this case. The helicopter is flying directly above the ocean liner. We can draw a vertical line between the helicopter and the ocean liner. Next we can draw a line from the helicopter to the other ship. The angle between this two lines and the angle of depression are complimentary 23˚ angles. Thus, the angle between 67˚ the two lines is 90˚ – 23˚ = 67˚. 1400 ft Now, drawing a line between the two ships creates a right triangle. The vertical height is the adjacent side of 67˚ and we are looking for the opposite side of 67˚. So, we will need to use the tangent function. x opp x tan 67˚ = = (multiply by 1400 to solve for x) adj 1400 x = 1400•tan 67˚= 1400•2.35… = 3298.1… ft ≈ 3300 ft So, the ships are approximately 3300 ft apart. € € a wheel Ex. 17 For chair accessible ramp to be ADA compliant, the ratio of the rise to the run of the ramp must be 1 to (12 or greater). A 29 ft 9 in long ramp leading into a building climbs 2 ft 5 in. a) Is this ramp ADA compliant? b) If a new city ordinance states that the incline of a handicapped access can be no more than 4.5˚, does this ramp comply with the new ordinance? If not, how long does the ramp need to be? Round to the nearest inch. Solution: a) We will begin by drawing a picture: Length ← Angle of Elevation Run Rise 164 The Length of the ramp corresponds to the hypotenuse, and the Rise and Run correspond to the two legs. Thus, we will need convert the measurements into inches and use the Pythagorean theorem to calculate the run of the ramp: ( 12in ) + 5 in = 24 in + 5 in = 29 in 1ft 29 ft 12in Length = 29 ft + 9 in = ( 1ft ) + 9 in = 348 in + 9 in = 357 in 1 2 ft 1 Rise = 2 ft + 5 in = (Run)2 = (Length)2 – (Rise)2 = (357)2 – (29)2 = 127449 – 841 = 126608 € € Run = 126608 = 355.52… in The ratio of the € Rise € to the Run is 29 to 355.52… Dividing by 29, we get: 1 to 12.269… Since 12.269… ≥ 12, the ramp is ADA compliant. € b) Since the ramp climbs 29 in, then 357 in x the height of the ramp is 29 in. This 29 in will correspond to the opposite side of the incline. The length of the ramp is 357 in. This will correspond to the hypotenuse. This means we will need to use the inverse sine function to find the angle. opp 29 sin x˚ = = (use the inverse sine) x= hyp 357 29 sin – 1 ( ) = 4.659…˚ 357 which is greater than 4.5˚ Thus, the current ramp does not comply with the ordinance. € climbs 29 in, then Since the ramp y the height of the ramp is 29 in. This 29 in 4.5˚ € will correspond to the opposite side of the incline. The length of the ramp is length of the ramp which is the hypotenuse. This means we will need to use the sine function. opp 29 sin 4.5˚ = = (multiply by y) hyp y y sin 4.5˚ = 29 (divide by sin 4.5˚) 29 = 369.619… ≈ 370 in y= sin 4.5 = 29 0.07845... € € But, 370 ÷ 12 = 30 with a remainder 10, so 370 in = 30 ft 10 in The € ramp€should be 30 ft 10 in to comply with the ordinance. Note: Another requirement for a ramp to ADA compliant is that is one 5 ft by 5 ft rest platform is required after a run of 30 feet in a ramp.