Warm-Up #32 Tuesday, 5/10/2016 Solve for x and find all of the missing angles. In triangle JKL, JK=15, JM = 5, LK = 13, and PK = 9. Determine whether line segment JL is parallel to line segment MP. Justify your answer.

Homework Area of Regular Polygons_ finish

Basic Trigonometry

The Trigonometric Functions SINE COSINE TANGENT

Represents an unknown angle Greek Letter q Prounounced “theta” Represents an unknown angle

Right Triangle Trigonometry The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides refer to another angle, other than the 90o. A

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.

hypotenuse hypotenuse opposite opposite adjacent adjacent

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 SOHCAHTOA is pronounced “Sew Caw Toe A” and it means 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 https://www.youtube.com/watch?v=PIWJo5uK3Fo

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

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

Finding an angle from a triangle To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle. We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio. 14 cm 6 cm C 1. Find angle C Identify/label the names of the sides. b) Choose the ratio that contains BOTH of the letters.

14 cm 6 cm C 1. We have been given the adjacent and hypotenuse so we use COSINE: Cos A = h a Cos A = Cos C = Cos C = 0.4286 C = cos-1 (0.4286) C = 64.6o

Find angle x 2. 8 cm 3 cm x Given adj and opp need to use tan: Tan A = a o Tan A = Tan x = Tan x = 2.6667 x = tan-1 (2.6667) x = 69.4o

3. 12 cm 10 cm y Given opp and hyp need to use sin: Sin A = sin A = sin x = sin x = 0.8333 x = sin-1 (0.8333) x = 56.4o

Finding a side from a triangle 7 cm k 30o 4. We have been given the adj and hyp so we use COSINE: Cos A = Cos A = Cos 30 = Cos 30 x 7 = k 6.1 cm = k

4 cm r 50o 5. We have been given the opp and adj so we use TAN: Tan A = Tan A = Tan 50 = Tan 50 x 4 = r 4.8 cm = r

12 cm k 25o 6. We have been given the opp and hyp so we use SINE: Sin A = sin A = sin 25 = Sin 25 x 12 = k 5.1 cm = k

x = x 5 cm 30o 1. Cos A = Cos 30 = x = 5.8 cm 4 cm r 50o 2. Tan 50 x 4 = r 4.8 cm = r Tan A = Tan 50 = 3. 12 cm 10 cm y y = sin-1 (0.8333) y = 56.4o sin A = sin y = sin y = 0.8333

Applications Involving Right Triangles A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.3. How tall is the Washington Monument? Solution: where x = 115 and y is the height of the monument. So, the height of the Washington Monument is y = x tan 78.3  115(4.82882)  555 feet.

Ex. A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree? tan 71.5° ? tan 71.5° 71.5° y = 50 (tan 71.5°) 50 y = 50 (2.98868)

Ex. 5 A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge? cos 60° x (cos 60°) = 200 200 60° x x X = 400 yards