Basic Trigonometry

Warm Up: What do you remember about similar triangles? C M N A T

Parts of a Right Triangle The hypotenuse will always be the longest side, and opposite from the right angle. Imagine that you are at Angle A looking into the triangle. The opposite side is the side that is on the opposite side of the triangle from Angle A. The adjacent side is the side next to Angle A.

Parts of a Right Triangle Now imagine that you move from Angle A to Angle B. From Angle B the opposite side is the side that is on the opposite side of the triangle. From Angle B the adjacent side is the side next to Angle B.

Review B A B A For Angle A Hypotenuse This is the Opposite Side This is the Adjacent Side A Adjacent Side Adjacent Side B For Angle B Hypotenuse This is the Opposite Side This is the Adjacent Side A

Trig Ratios Use Angle B to name the sides Using Angle A to name the sides We can use the lengths of the sides of a right triangle to form ratios. There are 3 different ratios that we can make. The ratios are still the same as before!!

Trig Ratios Each of the 3 ratios has a name Hypotenuse Each of the 3 ratios has a name The names also refer to an angle Opposite A Adjacent

Trig Ratios If the angle changes from A to B Hypotenuse If the angle changes from A to B Opposite A The way the ratios are made is the same Adjacent

SOHCAHTOA B Hypotenuse Here is a way to remember how to make the 3 basic Trig Ratios Opposite A Adjacent 1) Identify the Opposite and Adjacent sides for the appropriate angle SOHCAHTOA is pronounced “Sew Caw Toe A” and it means Sin is Opposite over Hypotenuse, Cos is Adjacent over Hypotenuse, and Tan is Opposite over Adjacent Put the underlined letters to make SOH-CAH-TOA

Similar Triangles and Trig Ratios Triangles are similar if the ratios of the lengths of the corresponding side are the same. Triangles are similar if they have the same angles All similar triangles have the same trig ratios for corresponding angles

Similar Triangles and Trig Ratios P B 5 20 3 12 A C 4 Q R 16 Ratio of corresponding sides:   They are similar triangles, since ratios of corresponding sides are the same Let’s look at the 3 basic Trig ratios for these 2 triangles Notice that these ratios are equivalent!!

Examples of Trig Ratios First we will find the Sine, Cosine and Tangent ratios for Angle P. 20 12 Adjacent Next we will find the Sine, Cosine, and Tangent ratios for Angle Q Q 16 Opposite Remember SohCahToa

Example: Finding Trigonometric Ratios Write the trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. sin J Note: When the problem states Find Sin J it is telling you two things! Go to angle J as the reference angle Find the ratio of the sides opposite/hypotenuse

Calculator Tip On a calculator, the trig functions are abbreviated as follows: sine  sin, cosine  cos, tangent  tan In trigonometry, the letter of the vertex of the angle is often used to represent the measure of that angle. For example, the sine of A is written as sin A. Writing Math

Example : Calculating Trigonometric Ratios Use your calculator to find the trigonometric ratio. Round to the nearest hundredth. sin 52° Be sure your calculator is in degree mode, not radian mode. Caution! sin 52°  0.79

Example : Calculating Trigonometric Ratios Use your calculator to find the trigonometric ratio. Round to the nearest hundredth. cos 19° cos 19°  0.95

Example : Calculating Trigonometric Ratios Use your calculator to find the trigonometric ratio. Round to the nearest hundredth. tan 65° tan 65°  2.14

Example : Using Trigonometric Ratios to Find Lengths Find the length of BC. Round to the nearest hundredth. Write a trigonometric ratio. Substitute the given values. Multiply both sides by BC and divide by tan 15°. BC  38.07 ft Simplify the expression.