How do we analyze the relationships between the angles and sides of right triangles? Agenda: Warmup Notes/practice – sin/cos/tan Core Assessment 1 Monday Right Triangles TEST Tuesday

Today’s objective is to set up trigonometric ratios. So far this unit we have found side lengths in right triangles by either using the _________________ or by using the ______________________ Pythagorean Theorem special triangle relationships. Now we can find out all information about a right triangle (all side lengths and all angle measures) if we know: a) one side length and one angle or b) two side lengths. We use trigonometric ratios to accomplish this goal. Today’s objective is to set up trigonometric ratios.

Hypotenuse Adjacent Opposite Trigonometric functions are always set up based on two of the three sides of a right triangle. The sides are called Opposite, Adjacent, and Hypotenuse. x° Adjacent Hypotenuse Opposite

Hypotenuse Adjacent Opposite To set up the sine ratio, you compare the opposite side to the hypotenuse. x° Adjacent Hypotenuse Opposite

Hypotenuse Adjacent Opposite To set up the cosine ratio, you compare the adjacent side to the hypotenuse. x° Adjacent Hypotenuse Opposite

Hypotenuse Adjacent Opposite To set up the tangent ratio, you compare the opposite side to the adjacent side. x° Adjacent Hypotenuse Opposite

There is an acronym to remember how to set up the trigonometric ratios: SOHCAHTOA S C O T A O H H A

Example 1 – Set up the 6 trig ratios for x and y. 5 4 y° 3 SOHCAHTOA

Ex 2 – Set up the 6 trig ratios for 20 and ___. 70° y 70° 4 20° SOHCAHTOA x

First use Pythagorean Theorem to find the missing side length. Ex3 – Set up the 6 trig ratios for x and y. SOHCAHTOA First use Pythagorean Theorem to find the missing side length. 52 + z2 = 132 z2 = 144 z = 12 25 + z2 = 169 x° 13 5 y° 12

We will look at two types of problems: 1) If you are given a right triangle with one angle measurement and one side length, then you can find the other two sides and last angle measurement. 2) If you are given a right triangle with two side lengths, then you can find the other side length and the two angle measurement.

SOHCAHTOA Type 1 – Example 1 x 10 y sin(55) = cos(55) = tan(55) = Step 2 – Set up the 6 possible trig ratios. sin(55) = cos(55) = tan(55) = sin(35) = cos(35) = tan(35) = Step 1 – Find the last angle measure. 35° x 10 55° y Step 3 – Choose two trig functions to use and solve for all variables. SOHCAHTOA Step 4 – Check your answers using the Pythagorean Theorem.

SOHCAHTOA Type 1 – Example 2 x y 10 sin(53) = cos(53) = tan(53) = Step 2 – Set up the 6 possible trig ratios. sin(53) = cos(53) = tan(53) = sin(37) = cos(37) = tan(37) = Step 1 – Find the last angle measure. 10 53° x 37° y Step 3 – Choose two trig functions to use and solve for all variables. SOHCAHTOA Step 4 – Check your answers using the Pythagorean Theorem.

SOHCAHTOA Type 1 – Example 3 y 9 x sin(59) = cos(59) = tan(59) = Step 2 – Set up the 6 possible trig ratios. sin(59) = cos(59) = tan(59) = sin(31) = cos(31) = tan(31) = Step 1 – Find the last angle measure. x 59° y 31° 9 Step 3 – Choose two trig functions to use and solve for all variables. SOHCAHTOA Step 4 – Check your answers using the Pythagorean Theorem.

Type 1 – Example 3 - Revisited 30° 60° 90° n 2n x 60° y 30° 9 y: x: