2 Objectives/Assignment Find the since, the cosine, and the tangent of an acute triangle.Use trionometric ratios to solve real-life problems, such as estimating the height of a tree or flagpole.To solve real-life problems such as in finding the height of a water slide.Assignment: pp #3-38 allDue TODAY– 9.4Quiz next time we meet
3 Finding Trig RatiosA trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan respectively.
4 Trigonometric RatiosLet ∆ABC be a right triangle. The since, the cosine, and the tangent of the acute angle A are defined as follows.Side adjacent to Abcos A ==hypotenusecSide opposite Aasin A ==hypotenusecSide opposite Aatan A ==Side adjacent to Ab
5 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.
6 Ex. 1: Finding Trig Ratios Compare the sine, the cosine, and the tangent ratios for A in each triangle beside.By the SSS Similarity Theorem, the triangles are similar. Their corresponding sides are in proportion which implies that the trigonometric ratios for A in each triangle are the same.
7 Ex. 1: Finding Trig Ratios LargeSmallopposite8sin A =≈0.47064≈0.4706hypotenuse178.5adjacent7.5cosA =15≈0.8824≈0.8824hypotenuse8.517oppositetanA =84≈0.5333≈0.5333adjacent157.5Trig ratios are often expressed as decimal approximations.
9 Ex. 2: Finding Trig Ratios—Find the sine, the cosine, and the tangent of the indicated angle. opposite12sin S =≈0.9231hypotenuse13adjacentcosS =5≈0.3846hypotenuse13oppositetanS =12≈2.4adjacent5
10 Ex. 3: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 45 opposite1√2sin 45==≈0.7071hypotenuse√22adjacent1√2cos 45==≈0.7071hypotenuse√22opposite1tan 45=adjacent=11Begin by sketching a 45-45-90 triangle. Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. From Theorem 9.8 on page 551, it follows that the length of the hypotenuse is √2.√245
11 Ex. 4: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 30 opposite1sin 30==0.5hypotenuse2adjacent√3cos 30=≈0.8660hypotenuse2opposite1√3tan 30==adjacent≈0.5774√33Begin 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 2.30√3
12 Ex: 5 Using a CalculatorYou 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.
14 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.
15 Trigonometric Identities A trigonometric identity is an equation involving trigonometric ratios that is true for all acute triangles. You are asked to prove the following identities in Exercises 47 and 52.(sin A)2 + (cos A)2 = 1sin Atan A =cos A
16 Using Trigonometric Ratios in Real-life Suppose you stand and look up at a point in the distance. Maybe you are looking up at the top of a tree as in Example 6. The angle that your line of sight makes with a line drawn horizontally is called angle of elevation.
17 Ex. 6: Indirect Measurement You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of the tree. You measure the angle of elevation from a point on the ground to the top of the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet.
18 The math The tree is about 76 feet tall. Write the ratio tan 59° =oppositeadjacentWrite the ratiotan 59° =h45Substitute valuesMultiply each side by 4545 tan 59° = hUse a calculator or table to find tan 59°45 (1.6643) ≈ hSimplify75.9 ≈ hThe tree is about 76 feet tall.
19 Ex. 7: Estimating Distance Escalators. The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg of 76 feet.30°
20 Now the mathsin 30° =oppositehypotenuse30°Write the ratio for sine of 30°sin 30° =76dSubstitute values.d sin 30° = 76Multiply each side by d.sin 30°76d =Divide each side by sin 30°0.576d =Substitute 0.5 for sin 30°d = 152SimplifyA person travels 152 feet on the escalator stairs.
21 Reminders: After this section, you have a quiz on Thursday or Friday. Chapter 9 exam will take place before you leave for spring break Take it before you go on break.Binder check before spring break. These are your new grades for the last quarter of the year. Study and don’t slack off now.