5-Minute Check

Section 7.5 Apply the Tangent ratio Students will analyze and apply the Tangent Ratio for indirect measurement. Why? So you can find the height of a roller coaster, as seen in Ex 32. Mastery is 80% or better on 5-minute checks and practice problems.

Finding Trig Ratios A 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.

Sample keystrokes Sample keystroke sequences Sample calculator display Rounded Approximation 74 0.961262695 0.9613 0.275637355 0.2756 3.487414444 3.4874 sin sin ENTER 74 COS COS ENTER 74 TAN TAN ENTER

Trigonometric Ratios – skill develop Let ∆ABC be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows. Side adjacent to A b cos A = = hypotenuse c Side opposite A a sin A = = hypotenuse c Side opposite A a tan A = = Side adjacent to A b

Ex 1-Find Tangent Ratios- skill develop

Think……ink…..share

Example 2-Find a leg length-skill develop

Ex 3….Estimate height using tangent real world application

Ex…4 – skill development

Ex 4 ….cont

Think….ink…share

What was the Objective for today? Students will analyze and apply the Tangent Ratio for indirect measurement. Why? So you can find the height of a roller coaster, as seen in Ex 32. Mastery is 80% or better on 5-minute checks and practice problems.

Homework PDF Tangent CFU

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.

Ex. 1: Finding Trig Ratios Large Small opposite 8 sin A = ≈ 0.4706 4 ≈ 0.4706 hypotenuse 17 8.5 adjacent 7.5 cosA = 15 ≈ 0.8824 ≈ 0.8824 hypotenuse 8.5 17 opposite tanA = 8 4 ≈ 0.5333 ≈ 0.5333 adjacent 15 7.5 Trig ratios are often expressed as decimal approximations.

Ex. 2: Finding Trig Ratios opposite 5 sin S = ≈ 0.3846 hypotenuse 13 adjacent cos S = 12 ≈ 0.9231 hypotenuse 13 opposite tanS = 5 ≈ 0.4167 adjacent 12

Ex. 2: Finding Trig Ratios—Find the sine, the cosine, and the tangent of the indicated angle. opposite 12 sin S = ≈ 0.9231 hypotenuse 13 adjacent cosS = 5 ≈ 0.3846 hypotenuse 13 opposite tanS = 12 ≈ 2.4 adjacent 5

Ex. 3: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 45 opposite 1 √2 sin 45= = ≈ 0.7071 hypotenuse √2 2 adjacent 1 √2 cos 45= = ≈ 0.7071 hypotenuse √2 2 opposite 1 tan 45= adjacent = 1 1 Begin 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. √2 45

Ex. 4: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 30 opposite 1 sin 30= = 0.5 hypotenuse 2 adjacent √3 cos 30= ≈ 0.8660 hypotenuse 2 opposite 1 √3 tan 30= = adjacent ≈ 0.5774 √3 3 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. 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.

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.

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 = 1 sin A tan A = cos A

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.

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.

The math The tree is about 76 feet tall. Write the ratio tan 59° = opposite adjacent Write the ratio tan 59° = h 45 Substitute values Multiply each side by 45 45 tan 59° = h Use a calculator or table to find tan 59° 45 (1.6643) ≈ h Simplify 75.9 ≈ h The tree is about 76 feet tall.

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°

Now the math sin 30° = opposite hypotenuse Write the ratio for sine of 30° sin 30° = 76 d Substitute values. 30° d sin 30° = 76 Multiply each side by d. sin 30° 76 d = Divide each side by sin 30° 0.5 76 d = Substitute 0.5 for sin 30° d = 152 Simplify A person travels 152 feet on the escalator stairs.

HW: Section 7.6 Page 477-478 1-27 all