Warm-Up 1 Find the value of x.

Warm-Up 1 Find the value of x.

History Lesson Right triangle trigonometry is the study of the relationship between the sides and angles of right triangles. These relationships can be used to make indirect measurements like those using similar triangles.

History Lesson Early mathematicians discovered trig by measuring the ratios of the sides of different right triangles. They noticed that when the ratio of the shorter leg to the longer leg was close to a specific number, then the angle opposite the shorter leg was close to a specific number.

Example 1 In every right triangle in which the ratio of the shorter leg to the longer leg is 3/5, the angle opposite the shorter leg measures close to 31 . What is a good approximation for x ?

Example 2 In every right triangle in which the ratio of the shorter leg to the longer leg is 9/10, the angle opposite the shorter leg measures close to 42 . What is a good approximation for y ?

Trig Ratios The previous examples worked because the triangles were similar since the angles were congruent. This means that the ratios of the sides are equal. In those cases we were using the tangent ratio. Here’s a list of the three you’ll have to know. sinecosinetangent

Trigonometric Ratios I Objectives: 1.To discover the three main trigonometric ratios 2.To use trig ratios to find the lengths of sides of right triangles

Investigation 1 Use the GSP Activity to discover the three main Trigonometric ratios sine, cosine, and tangent.

Summary hypotenuse side adjacent Θ side opposite Θ

Summary hypotenuse side adjacent Θ side opposite Θ

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Example 3 Find the values of the six trig ratios for α and β.

Activity: Trig Table On the previous example, we knew all the sides of the triangle, and we just listed the three trig ratios for those sides using a generic angle. Usually, though, you know the angle, and you want to find a side. Nowadays, we would use a calculator to find the sine or tangent of an angle. In the long, dark years before the calculator, people had to find their trig ratios in a table.

Activity: Trig Table In the 1500s, Georg Rheticus, a student of Copernicus, was the first to define the six trig functions in terms of right triangles. He was also the first to start a book of values for these ratios, accurate to ten decimal places to be used in astronomical calculations.

Activity: Trig Table Step 3: Set up a table of values like so: θsin θcos θtan θ 20° 70°

Activity: Trig Table Step 4: Now use your calculator to round each calculation to the nearest thousandths place. θsin θcos θtan θ 20° 70°

Activity: Trig Table Step 5: Finally, let’s check your values with those from the calculator. For sin, cos, and tan 1.Make sure your calculator is set to DEGREE in the MODE menu. 2.Use one of the 3 trig keys. Get in the habit of closing the parenthesis.

Example 4 To the nearest meter, find the height of a right triangle if one acute angle measures 35° and the adjacent side measures 24 m.

Example 5 To the nearest foot, find the length of the hypotenuse of a right triangle if one of the acute angles measures 20° and the opposite side measures 410 feet.

Example 6 Use a special right triangle to find the exact values of sin(45°) and cos(45°).

Example 8 Find the value of x to the nearest tenth. 1. x =2. x =3. x =