mcr3u unit 5 lesson notes - connell math /...

MCR3U Unit 5 Lesson Notes

1

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

Write the 6 trig ratios for angle A


2

Ex 2. 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 Trig Ratios: P.281 #1 3 4 5ab(ii iv) 6 7a 8a 1012 14 15 16b


3

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

ex. 2 Determine the exact value of

a. b.

ex. 3 Determine the exact value of

450

450 600 600

300


4

* 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

Ex. Draw the angles in standard position.

a) θ = 40 b) θ = 130 c) θ = 215 d) θ = 340

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

5.4 Evaluating Trigonometric Ratios (part 1)


5

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

Ex: 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°


6

Ex. Evaluate:

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

Ex. State the exact value: (hint: use the special triangles)

a) sin30 ° b) sin150° c) sin210° d) sin330°

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

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

Ex. Determine the values of θ, when 0 θ 360°

a) cosθ=0.8988 b) sinθ=-0.8290

5.4 Part 1 Assignment: p.299 #1, 35, 8, 10


7

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

where

Ex 1. The point (-9, 4) lies on the terminal arm of angle θ in standard position.a) Sketch angle θb) Determine the exact value of rc) Determine the primary trig ratiosd) Calculate θ

5.4 Evaluating Trigonometric Ratios (part 2)


8

Example 2: Given that and A lies in the first quadrant,

a) Determine the exact values for cosA and tanA.

b) Determine A

Example 3: 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, 6bcf, 12 p.304 #113 (midchapter review)


9

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:

→Reciprocal Trig Identities

→Quotient Trig Identities

→Pythagorean Trig Identities

Ex 1. Simplify

a. b.

Ex 2. Factor

a. b.

and and

and


10

Ex 3. 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

*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.


11

5.5 Part 2 Assignment: P.310 #8cd 10 11 12ab 14?

b) sin2θ = 1 + cosθ 1-cosθ

5.5 Trigonometric Identities Day 2

c)

d)


12

MCR3U 5.6 The Sine Law

Given triangle ABC, with CD drawn perpendicular to AB. Then CD is the height h of the triangle. The height now separates triangle ABC into two right triangles, CDA and CDB.

We could twist the above triangle around and repeat this process using the other side of the triangle to prove:

The Sine Law

Ex.1: Solve the following triangle.

A

B

C

7.9 cm

.


13

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

Ex.2: 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


14

5.7 Cosine Law and 3-Dimensional Problems

Recall: Cosine Law

Ex. Trevor sees a swimmer in distress on the other side of a river. He measures the angle of elevation from a rock on his side of the river to the top of the cliff above the swimmer to be 28 ̊ and measures an angle of 85 ̊ from the river bank to the swimmer. He walks 50m downstream and measures an angle of 32 ̊ from the river bank to the swimmer. Determine height of the cliff and the distance from the initial rock to the swimmer.

When do we use it?


15

Ex2. 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 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


16

5.8 3 Dimensional Trigonometry with Bearings

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.

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.

5.7 Practice Continued: P.333 #4a, 5, 8, 9, 11

mcr3u unit 5 lesson notes - connell math /...

Documents