MHF 4U1 UNIT W TRIGONOMETRIC FUNCTIONS

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MHF 4U1

UNIT W

TRIGONOMETRIC

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c) (1)

3. In Exercise 2, find two angles which are coterminal with 8.

4. An angle 0 in standard position is shown. Find two other angles tdhich are cotenninal with 0.

y

6. p is a point on the terminal arm of an angle 8 in standard rotated 4200.

a) How many complete rotations have been made? b) In which quadrant is P located now?

c) Draw a diagram to show the position of P. 7. Repeat Exerctce 6 if P has rotated:

a) 480° _{b) 660°}

8. Draw each angle in standard position.

position. Suppose P has

a) 0 = 400° b) 0 = 750° _{c) 0 = —270°} _{d) 0 = —60°}

9. Repeat Exercise 6 if P has rotated radians.

ID. Repeat Exercise 6 if P has rotated:

a) it _{c) 2ir}

d). radians?

a)

I. An angle in standard position is shown. What is the value of 0, in degrees and in b) y p x Iv y _y x 120

2. Draw each angle in standard position.

‘0 p a) 0 = 50° e) 0 = 1,) 0 = 120° f) 0 = 4 c) 0 = 165° 2i, d) 0 = 240° a) h) 0 = b) _c) _y x 5” 1 d) 2100

5. Find two angles which are coterminal with 8.

S a)01T S )1T I 0 p c) 0 — d)0 —2ir c) 8700 _d) _1000°.

II. Draw each angle in standard position.

a) 0= _{b) 8} _{c) 0}

—

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Exercises 8-2, page 310

1. a) 1800,11 _{b) 450°,}

c) —90°,

—

il) —270°,

_—

_{3. Typical answers:}

5n 3n 9ir 7iy 8,r 4n

T’T

0i’T

g)7,-7

7,r iv

Ii)

_—j—.

_—; _{4. Typical answers:} a) 420°, —300° b) 150°, _5700

I3ir 3rr c) —. ——

4 4

Sn 3n

5. Typical answers: _{a) 3n,} ₁₁ _Ii)

j—,

d)0,—4n 6.a)l b) Typical answers: 580°, 940°, —140° 10. a) 0.95 11 _{b) Typical answers: 7.53, 13.82, —5.03} c) n + 2nn 1)

—;

+ nit IL a) 80.71 h Typical answers: 10.98, 17.27, —1.58

T

+ 2n,, Ii) I + 2n,, — 3 l4.h)R;0°y<2n,y€R 12.b)P(3,2) _{c)sinO=—,cosO=——=} 13. b) P(—2,5) c) sin U = /59’ cost) = 14. b) P(— 1,21 c) cos U = —, _(anti ₌ _—2 15. a) sin ü , tan

a

= V5

v’13

h) cos B = ——--—, tan B ₌ 3 2 c) sin U = ——,cos U = VI] 3

tI) ens U = —‘tan U =

16. a) Typical answers: 30°, 150°, 390° b) An infinite number a) 410°, —310° c) 525°, —195° b) 480°, —240° d) 600°, —120° 371 Sn d) 2 ‘ — 2 Exercises 8-3, page314 1. a) 0.8000, 0.6000, 1.3333 b) 0.8000. —0.6000, —1.3333 c) —0.7071, —0.7071, 1.000 d) —0.923!, 0.3846, —2.4000 2. a) 0.1961, —0.9806, —0.2000 Ii) —0.8944, —0.4472, —2.000 c) —0.3162, 0.9487, —0.3333 d) 0.8934, —0.4472, —2.000 3. a) 0.766 h) —0.819 c) —1.192 ci) —0.342 e) —0.171 f) —0.574 g) —0.231 h) —0.995 4. a) 0.389 b) 0.825 c) 3.010 ci) 0.974 e) —0.990 1) 1.587 g) —0.779 Ii) —0.781 5. b) —0.6, 0.8, —0.75 6. b) 0.891, —0.447, —2.000 512 5 I 21 7.a)

_-—-m-j

b)-,--I 3 I 4 34 c) — ci) —. I 3 I 9 2 15’V3’45 g) 1,0, undefined _{Ii) 0, —1,0} 8. a) 0.819 15 b) Typical answers: 485°, 845°, —235° 9. a) —0.76603 b) First 7. a) c) 2; second ci) h) First 10. a) I; second b) I; fourth 2; fourth _{9. a) I}

0; between second and third Ii) _{0; between third and fourth}

c) I; between fourth and first ci) 1; between first and second 12. a) 45° ÷ 360°,, hI 150° + 360°,,

c) 240° + 360°,, d) —30° + 360°,,

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1. Solve for 0 to the nearest a) sin U 0.35 d) cos U = 0.8492 a) sin 0 = 0.82 d) cog 0 = 0.1123 degree, 0° B 90°. b) cos U = 0.112 e) sin U = 0.9044 b) cog 0 = 0.75 e) sin 0 = 0.2552 c) tan U = 0.485 f) tan B = 2.058 c) tan 0 = 0.685

0

tan 0 = 3.158

3. Solve for 0 to the nearest degree, 0° 0 360°.

a) sin 0 0.75 I) cos 0 = 0.0965

d) enS 0 0.3558 e) sin B = 0.6666

5. Solve for U to the nearest degree, 0° 0 360°.

a) sin 0 = —0.6855 b) cos U = —0.1881

c) sin 0 = 0.1392

1) cog 0 = 0.9676

c) sin 0 = —0.2550

8. The point P( —2,— 6) is on the terminal ann of an angle 0 in standard position.

Find a value of 0 to the nearest degree.

9. The point given is on the terminal arm of an angle 0 in standard position. Find a

value of 0 to the nearest degree.

a) P(— l,—4) b) Q(3,—4) c) R(2,—3) d) S(— 1,2)

12. Solve for B to the nearest tenth of a

a) sin 20 = 0.75

c) 3 sin22O — 10 sin 20 ± 3 = 0

0 360°.

b) 3 cos 0 ±

d) 3 sinU +

23. Write a property of the tangent function which is similar to the properties of sine

and cosine functions which were developed in this section. 14. Solve for 0 to the nearest

a) 3 tan 0 — 2 = 0

c) 2 tan2B — 3 tan 0 + I = 0

b) 2 tan B + 7 = 0

d) 3 ta&B — 2 tan 0 — 4 = 0

IS. Solve for 0 to the nearest tenth of a degree, 0° U 90°.

2. Solve for 0 in radians to 2 decimal places, 0 0 IT

10. Solve for B to the nearest degree, 0°

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7. Find each value of 0 to the

a) sin 0 = 0.906

d) cot 0 = —1.428

g) tan 0 —0.532

j)

csc 0 1.086

nearest degree if 0 is obtuse.

b) cos 0 = —0.574 c) tan 0 —3.732

e) csc 6 = 1.743 f) sec 0 = —2.669

h) sin 0 = 0.978 I) cot 0 —0.123

k) cos 0 = —0.777 1) sect) —1.010

8. Given 0 is an obtuse angle and the value of one trigonometric ratio, find the other trigonometric ratio, and 0 to the nearest degree.

a) sin6 = j; find sec Oand 0

c) cos 0 = —;findcot0and 0

e) tan 0 = —3; find csc Oand 0

3. Find each value of 0.

b) cot 0 =

--; find cos 0 and 0

d) csc U = ; find tan 0 and 0

f) sect) = —; find sin Oand6

45 a) tan 0 = I e) csc 0 = 2 V3 z) sin 0 = b) sin 0 0 f) cos 6 =

j)

csc 0 = c) sec 0 = 2 g) tan 6 = 0 K) cot 0 = d) cos (3 = h) cot 0 = I) sec 0 =

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State the value of each ratio.

a) sin 300 b) cos 600 c) sec 30’

g) sin 450 h) cot 600 i) cos 45°

2, State the value of each ratio

a) csc 30° b) tan 60° g) tan 90° h) cos 0° La) b)- c)

nvi

k)V2 Dy’1 d) tan 450 e) csc 60° f) cot 300

j)

tan 30° Ic) csc 450 _I) _sec 450

I. Use a calculator to find the value of each Irigonometric

ratio to 3 decimal places.

a) sin 98° b) Ian I 13° c) cos 124° d) sec 174°

e) csc 161° f) cot 143°

2. For each obtuse angle U, state the six trigonometric ratios.

Evaluate each trigonometric ratio to 2 decimal places.

a) cos 110 h) cot 95° c) csc 138° d) tao 108°

g) sec 1150 _{h) cot 130°} i) csc 65° j) cos 140°

15,8) 3,

1

e) Sec I 35° 1) sin 135° k)’csc 135° I) sec 120° e) sin 142° f) tan 170°

Ic) sec 175° I) sin 100°

c) sin 0° d) sec 90° e) cos 30° f) cot 45°

i) sec 60°

j)

csc 0° Ic) cot 0° 1) sin 90°

ifli I’) •i) j) 2. a) 2 b)

v)

) 0 d) e)— I) I g) Ii) I 1)2 2 j) k)°° 1)1 h) a) Pç—5,12)

I

—4,3) x c) —

Or

3. Evaluate each trigonometric ratio. Give

a) cos 120° N sin 150° c) cot

350

g) tan 120° hi cot 120° ii sin 120°

0 0 x exact answers. d) csc 150°

j)

cot 150° t ,fl ii ‘0 0 Li 1 . . > t _IrI° I u,Ifl vijDin(fl I 2oc eqfl I. I .,., U . .. tIcn U II eq I U —% r i°’ C 9 rIrt ir_n — — I ‘‘I I . —— C r4eq I I r-O I U I no II - Ct I 0 . .IflI I u ‘0 — n1 — cc C’—

—I—

U ‘nIn U

—I°

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WORKSHEET #13: FINDING EXACT VALUES OF ANGLES N RADIANS

1. Find the exact values of the following:

.t K K

a)

sm—

b)

cos—

c)

tan—

4 ₃

₆

ci)

sin

e)

co4

_{Q tan}

g)

csc-

h)

sec

i)

cot-f

K K if

j) csc—

k) sec—

1)

cot—

3

6

4 2. Find the exact value of the following:

.5K

_7z

_5,r

a)

sm—

b) cos—

c)

tan—

3

4

6 .11K

4K

_5,r

ci)

sm—

e)

cos—

tan—

6 ₃

₄

7K

- 2K

g)

sin—

h)

cos—

i)

tan—

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WORKSHEET #13B: FP1DNG EXACT VALUES OF ANGLES N

RADJANS

3. Find the exact value of the following:

• —2,r

_-3g

_-7ff

a) sm

b) cos

c)

tan

3

4

6 • —Sn.

-5g

—4,r

d)sm

e) cos

tan

4

6 ₃

• —11,r

_-5n.

.

-Li

g)

sin

_h)

cos

_tan

6

3

4 4. Find the exact value of the following:

• 9z

liz

19z

a) sin—

b) cos—

c)

tan—

4

3

6 • IOn.

17z

15z

d) sin

e) cos

1’)

tan—

3 ₆

₄

5. Find the exact value of the following:

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MHF 4U1

_{TRIGONOMETRIC GRAPHS}

La/S

.1u(<

For each of the following trigonometric functions:

i) List the key features

ii) List the transformations/translations

iii) Graph (1 or 2 cycles) the fimctions

and

the

transformationslftanslations

_needed.

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WORKSHEET #14: FINDING EXACT

VALVES

OF ANGLES N

RADIANS

USING TilE SUM AND DIFFERENCE FORMULAE

1. Find the exact values of the following:

5,r

7n

a)

sm—

b)

cos—

e)

tan—

12

12 • Ilir

13z

17,r

d) sin—

e)

cos—

f)

tan-jj

•19g

23K

g) sin—

h)

cos

i)

tan—

12

5K

_7K

_ilK

j)

csc—

_Ic)

sec—

_{I) cot—}

12

12 ₁₂

l7K

19z

23K

j)

csc—

_Ic)

sec—

_I)

_cot

12

12 2. Find the exact value of the following:

• (—7,r’

(—17K’

(—5,r

a)

smi

I

b) cosi

I

c)

tani

k12)

₁₂₎

₁₂

• (—i1,r

_(—5z’

_—19g

d) sm

I

e)

cost

I

k

12)

12

___

3IK

35K

g)

sin

_h)

_cos—

_{1) tan}

12

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B 1.

Express each of the following as a function of its related acute angle

and evaluate.

/ 7ir

15w

(a) sinl——)

(b) cos

—

F 6 4

8ir 33ir

(c) tan(—-j-)

(d) tan

(e)

sin 240°

(f) cos(- 135°)

(g) tan 330°

(h) sin 495°

2. Express each of the following as a function of its co-related acule

angle and evaluate.

11

.,

/

7’u

(a) cos

--it

(b)

(c) sin 120°

(d) tan(—f)

(e) tan 510°

(f)

cos(—315°)

3. Simplify.

ens

r + ros(r.—

x)

—

cos(

+

x)

—

cos(—x)

(I,)

tan x

+

tanQn

—

x)

+

cot(

— —

tan(2ir

—

x)

(c) siI@+x)+cos—x)+tan(+x)+tan(2E_x)

11 .3’n

(d) sin(-

+

—

cos(7

—

+

sIn(-j-

—

x)

(e) sin

—

x)

+

sin(it

—

x) + sinQ1

—

+

sin(2it

—

x)

4,

Find the cosecant, secant, and cotangent of each of the following.

Express your answers

iii

terms of cosecant, secant, or cotangent of x.

(a)

it—s

(b) +x

(c)

it+x

_(d)

5. Simplify.

(a)

sine

—

it)

(b) cos(x

—

(c) tan(—x

—

it)

6.. Evaluate.

(a)

secQw

±

(b)

—

(c)

_cot(!+

)

(d)

(e) cscH

—

(f) cot

(—

+)

7. Simplify.

cos(it

+

x)cos(

+

)

sin(ç

—

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b 2. _{hvaluate using formulas developed in this section.} Hit _13r (a) sin

_(b)

ens

—w

(c) tan(—-r) _(d) _{tan(_-j31r)}

(e) sin 75° _(f) _{cos(— 15°)}

3. _{Find the value of each of}

_the

_following.

.fr

rh

fit r\ _/ 3r lit (a) sinu —

)

(b) cos--

)

(c) tan\-7 +

fr\

.

-4. _{If x andy are in}

_the

_interval

ç0 —)

and sin x = and cos y

=

evaluate each of the following.

(a) sin(x

—

y) (b) cos(x ₊ _(c) _tan(x

* y)

fir _\ ‘. . .

/

3r\

i x is in the interval —,-iv) andy is in the interval l\r

-i)

and cos x

=

—

and tan y = evaluate each of the following.

(a) _{sin(x + y)} _{(b) cos(x} _y)

(c) tan(x — y)

6. _{Find the exact value of each of}

_the

following.

(a) _{sin 50° cos 20°}

— cos 50° sin 20° it 4’rr

.

(b) cos-cos

j—

—

sIn-sinj-j

tan 7° + [an 8° (c)

I— tan 7° tan 8°

Sri

5-ri 5h5r

(d) sin cos

-j-

+ cos - sin

10. _{Prove each of the lollowing.}

(a) sin(it + x) =

—

sin x _(l) _tan(2r

—

_x)

= —tan x

(c) cos(f + = sin x _(d) _sinç

— =

—

(e) cos(1 +

x) sin x (f) tanL + =

—

cot x

(g)

sin(x

—

it) = —sins _(h) _—tan(—.x

—

-iv) = tan x

21. Simplify.

3-ri /3-ri

(a) cos(-— + + sç4

—

x

(I,) cos( — sec

—

sinH — sin(x — + — + sin(y

—z)

(c)

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WORKSHEET #16: FINDING EXACT

VALUES

OF ANGLES EN RADIANS

USING THE DOUBLE ANGLE FORMULAE

1. Find the exact values of the following:

.3r

lix

a)

sm—

b)

cos—

c) tan—

8

12

24 • 19,r

5ir

13x

d)

sm—

e)

cos—

_{0 cos—}

24

8

24 52r

5,r

hit

g)

tan—

h)

see—

ese—

8

24

24 3,r

•7ir

5g

j)

cos—

k) sm—

I) tan—

16

16 41it

9ir

29z

m)

cos—

n)

sin—

o) cos

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5. Solve for 0 to the nearest degree, 00 ₀ _360°. 11

li’

6 ‘3

ii) in radians to 3 decimal places.

a) sin 0 = 0.7295 _{b) cos 0} = —0.3862 _{c) tan B} = —5.1730

6. Solve for 0 in radians to 2 decimal places, 0 0 2m

7. Solve for 0 to the nearest degree, 0° 0 360°.

a) 3sinU+ 2=0 _{b) 2tanO—5 =2}

c) 12 sin2O — II sin B + 2 = 0

e) 3 cos2U + 2 = 4

8. Draw graphs o y = sin 0 and v = cos 0 for —360° ₀ 360°. For each graph

a) State the maxiinuni value of v, and the values of 0 for which it occurs.

b) State the minimum value of y, _{and the values of 0 for which it occurs.}

c) State the 0- and y-intercepts.

-9. Find the amplitude, the period, the phase shift, and the vertical displacement for each function.

a) y = 3 sin 2(0 — 45°) — 4 b) y = —2 cos 5(0

+ + I

10. Sketch (lie graphs of each set of functions on the same grid for —2 0 2u.

with it. a) 65° e)

1. Draw each angle in standard position, then find two angles which are coterminal

b) 135° _{c) 200°} _d) _—450°

g)

2. Determine the sine, the cosine, and the tangent to 3 decimal places of each angle in Exercise 1.

3. Each point P is On the terminal arm of angIe 0. Find sin 0, cos 0. and tan 0 to 3 decimal places.

a) P(49) b) P(8,—15)

4. Find each value of 0 in Exercise 3:

/ i) in degrees to 1 decimal place

c) P(—4,7) d) P(—6,—5)

a) cos 0 = 0.2681 _{b) tan 0} = 1.0744 _{c) sin 0} = —0.4683

d) 3 cos2O + 4 cos 0 — 2 = 0 f) 2 sin1O ± 5 sin 0 + I = 0

a)ysino y=3sinf) y3sinO+2

b)vcos0 vcos(+;) y=cos(of)

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7. Angle measures Write each angle in degrees.

a)

Irrad

b)

rad

tad

cad

e) rad

f)

irad

8.

Angle measures Write each angle in radians.

a)

b) 750

c)

105° d) 120°

e)

f5QD f) 700°

9. Trigonometric functions Determine the

values of sin 0, cos 0, and tan

0.

______

___-10.

Trigonometric functions Use appropriate

right-angled triangles to determine exact values

for the following quantities.

30° or

-‘

;4LL

___J

60° or

—

-1 4

3

Ii.

Trigonometric functions Use a calculator to

determine the sine, cosine, and tangent of the

following angles, rounded to four decimal

places.

a)

27.7°

b) 81.4°

c) 0.8 tad

IT d) -

rad

12. Trigonometric functions Use your calculator

to determine an angle, in radians, rounded to

two decimal places, for which

a) sinO=0.34

b)

cosO=0.6

c)

tanO=4

17. Determining angles For each diagram.

determine the angle

0,

in radians

a)rJVJiJIb)iif\iifT) __

iN4I

H

MHITI L

itiH fl

c)l

I

_J

18.

Graphing trigonometric functions Sketch a

graph of two cycles of each function

a) =smx

b)

=cosa

c) y=tant

I d) ,‘=sin(r—

)

e) y=3cos(2)

25.

_{Trigonometric functions Determine the exact}

ialue of each quantin

a) csc

b)

sec

c)

d)

secO

e)

csc0

f)

corn

a) csc3o°

h)

sec 90°

i)

_{cot 60°}

7 a) 180° Ii) 90°c)60°ii) 45°c) 30° 1)5 296° 8a) b) c) d) 2ir c) I) 434 I 2 1 f_I

“

ga)_, _._b)r_._5lOSiflO

‘‘_T’

cusP ‘, , , tin 0 , I, 11 a) U1648. 08854,0 525Db) 09888,01495,66122 c)07174, 0696Th I o:96d)04339,o9010,04SI6 12 a) 035 b)093 c) 133 13 14 a) Hint AD=5mb, OR =cosOb) No JO812+ (02)’ 116 a) 132139. 38302)b) (-2 2943 32766) c) (—2 2981,—I 9284’ d) (3,

—311)

17 a)0 588 rid b) 2 2143 ridc) 36052 nu d)c42sridlsa;o,2n-bç r—,’c)I5 413,530 25 a)

,,7=

Ii)

J7

c) I d) I C) undctintd

c)

d)

[5°

ri

d)F

Li L

a)J I 2

I) undefinedg)2 ii) undefined I)