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5.1 Trigonometric Ratios in Right Triangles

3 Primary Trig Ratios: Reciprocal Trig Ratios

Remember : SOH CAH TOA

• To solve a triangle means to determine all unknown side lengths and angles

A

BC 3

4

Ex1 Write the 6 trig ratios for angle A

Ex2 Solve for angle A

Ex3 Determine θ if,

tan θ = 1.6 cos θ = 1.6

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Ex4 Solve the triangle.

25 15

C

B

A

Ex 3 Evaluate

a. sin 30o b. tan 60o

c. sec 40o d. cot 70o

e. f.

5.1 Assignment: P.281 #1, 3, 4, 5ab(ii iv), 6, 7a, 8a, 10­12, 14, 15, 16b

Ex5 Evaluate

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5.2 Evaluating Trig Ratios for Special Triangles 45, 30, 60When we use a calculator to determine trigonometric function values, we are approximating up to 9 decimal places.

However, for some special angles, exact values can be determined from geometric relationships.

Ex1 Determine the exact value of

Ex2 Determine the exact value of

a. b.

Ex3 Determine the exact value of

450

450 600 600

300

Ex4 Determine the exact length of x

x12

Ex5 Determine the exact length of x

x

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* An angle in standard position : The position of an angle rotating about the origin and its initial arm is the positive x-axis.

Exploring Trigonometric Ratios

Recall: The equation of a circle with its center at the origin is: 

The Unit Circle : center is at O(0, 0), and has a radius is 1 unit in length equation is :

Trig Ratios and the Unit Circle: OP is called the terminal arm, OA is the initial arm θ is the angle of rotation measured from the initial arm to the terminal arm. ∆ PON is a right triangle

Therefore, the coordinates of any point (x, y) on a unit circle are related to θ such that :

(x, y) = (cosθ, sinθ) and

Ex1 Draw the angles in standard position. 

5.2 Assignment: Page 286 #3, 4, 5a, 6b, 7, 11, 12 

5.3 Evaluating Trigonometric Ratios (part 1)

a) θ = 40 ̊ b) θ = 130 ̊ c) θ = 215 ̊ d) θ = 340 ̊           e) θ = -65 ̊ 

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As we rotate around the unit circle, the signs of the different trigonometric ratios change.

Quadrant II: x is negative, y is positive Quadrant I: x is positive and y is positive

Quadrant III: x is negative, y is negative Quadrant IV: x is positive, y is negative

I

(x,y)

All + 've

II

(-x,y)

Sine + 've

III

(-x,-y)

Tan + 've

IV

(x,-y)

Cos + 've

We can remember the sign of each trigonometric function in each quadrant by using the CAST rule.

C cos θ is positive in the 4th quadrant

A all trig ratios are positive in the 1st quadrant

S sin θ is positive in the 2nd quadrant

T tan θ is positive in the 3rd quadrant C

AS

T

Ex2 State whether each trig ratio is positive or negative without using a calculator. Then check with a calculator.

a) sin110 ° b) cos205° c) tan340° d) cos(-22°) e) tan79°

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Ex3 Evaluate (using your calculator):

a) cos20 ° b) cos160° c) cos200° d) cos340°

Ex4 State the exact value: (hint: use the special triangles)

a) sin30 ° b) sin210° c) sin135° d) sin300°

Ex5 State an equivalent expression in terms of the related acute angle:

a) sin130 ° b) tan150° c) cos300° d) sin250°

Ex6 Determine the values of θ, when 0 θ 360°

a) cosθ=0.8988 b) sinθ=-0.8290

5.3 Part 1 Assignment: p.299 #1, 4, 5, 8

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Recall: If (x, y) is any point on the terminal arm of a circle, the trig ratios can be determined as follows:

Ex1 The point (-9, 4) lies on the terminal arm of angle θ in standard position.a) Sketch angle θb) Determine the exact value of r

c) Determine the primary trig ratios

d) Calculate θ

Ex2 Determine the primary trig ratios for 900. Draw a diagram to help.

5.4 Evaluating Trigonometric Ratios (part 2)

SYR CXR TYXSOH CAH TOA

Remember:  or

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Ex3 Given that and ∠A lies in the first quadrant,

a) Determine the exact values for cosA and tanA.

b) Determine ∠A

Ex4 Given that and θ lies in the second quadrant,

a) Determine the exact values for sinθ and cosθ

b) Determine θ

5.4 Part 2 Assignment:  p.300 #2bd, 3, 6bcf, 10, 12

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5.5 Trigonometric Identities

Identity: An equation that is always true regardless of the value of the variable. ex: 5x = 2x + 3x

A trigonometric identity is a relation among trig ratios that is true for all angles for which both sides are defined.

The Basic Trig Identities (need to know these!!!):

→Reciprocal Trig Identities

and and

and

Ex1 Simplify.

a. b.

Ex2 Factor, then simplify, if possible.

a. b. c.

→Quotient Trig Identities

→Pythagorean Trig Identities

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Ex3 Prove:

a.

b.

5.5 Part 1 Assignment: P.310 #2, 3, 5, 7, 8ab 

TIPS for Proving Trig IDENTITIES- do NOT change to x,y,r- try to change all to either sin0 or cos0- work on sides alone LS or RS (do not bring things over to the other side)- look for factoring possibilities- look for common denominators to add/subtract terms- start with more complicated side- always look to the other side to see your "goal"

*We can use the basic trig identities to prove more complex trig identities. * To prove an identity, we show that the left side of the equation equals the right side of the equation.

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5.5 Part 2 Assignment: P.310 #8cd, 10, 11, 12ab

b) sin2θ = 1 + cosθ 1-cosθ

5.5 Trigonometric Identities Day 2

c)

d)

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5.6 The Sine Law

Ex1 Solve the following triangle.

A

B

C

7.9 cm

.

The Sine Law

Solving Triangles:

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The Ambiguous Case of the Sine Law: Ambiguous means not clear

Ex2 Solve triangle ABC with , a = 1.5, and b = 2. There are actually 2 ways we could sketch this:

This only happens when given SSA (side, side, angle) and when the side opposite the given angle (side a here) is smaller than the adjacent side (side b here) and larger than the height from point C drawn perpendicular to side AB or side c. From the given info, it isn't clear which triangle we have, so we must solve both!

5.6 Assignment: P.318 #4ab, 5ab, 6, 7, 8, 9, 12

C

A B

b a 

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5.7 Cosine Law and 3-Dimensional Problems

Recall: Cosine Law

Ex2 Trevor wants to measure the height of the cliff, but he thinks it is impossible, as there is a fast moving river below it. Sheri tells him it isn't impossible, he just needs to do some measurements and use some trigonometry.

Sheri looks directly across the river and sees a rock at the base of the cliff. She measures the angle of elevation from her side of the river to the top of the cliff to be 28 .̊ She also measures an angle of 85  ̊from the river bank to her line of sight directly across the river to the rock. She walks 50m downstream and measures an angle of 32  ̊from the river bank to the same rock at the base of the cliff.

She says she now has enough information to determine the height of the cliff using trigonometry. Is this true? Justify your answer.Has she made any assumptions?

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Ex3 A helicopter is hovering 40m above the ground. The pilot sees a hunter and a deer some distance apart. He determines the angles of depression to the hunter and deer to be 8 ̊ and 13 ̊ respectively. From the pilot, the angle between the line of sight to the hunter and deer is 117  ̊. How far apart are the hunter and deer?

5.7/5.8 Assignment: P.325 #4cd, 5, 6, 9, 10 P.333# 3bd

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5.8 3 Dimensional Trigonometry with Bearings

Ex1 A ship travels from Port A to Port B, 60 km away on a bearing of 35

a. Draw a diagram showing this information

b. If the ship returns to Port A using the same path, determine the bearing from B to A.

Ex2 A boat sails from A on a bearing of 140 for 40km to B. then sails on a bearing of 250 for 80km to C.

a. Find the distance from A to C. b. Find the bearing of C from A.

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Ex3 A spotter in a rescue helicopter 350m above the sea spots a drifting yacht at a bearing of 248 degrees and at an angle of depression of 27 degrees. The spotter also notes a reef with a bearing 191 degrees and angle of depression of 12 degrees. How far is the yacht from the reef?

5.8 Bearings Assignment: P.327 #8, 14 p.333 #4a 6 9

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