C1: Chapters 8 & 10 Trigonometry

Dr J Frost (jfrost@tiffin.kingston.sch.uk)www.drfrostmaths.com

Last modified: 1st September 2015

Sin GraphWhat does it look like?

90 180 270 360-90-180-270-360 ?

Sin GraphWhat do the following graphs look like?

90 180 270 360-90-180-270-360

Suppose we know that sin(30) = 0.5. By thinking about symmetry in the graph, how could we work out:

sin(150) = 0.5 sin(-30) = -0.5 sin(210) = -0.5 ? ? ?

Cos GraphWhat do the following graphs look like?

90 180 270 360-90-180-270-360 ?

Cos GraphWhat does it look like?

90 180 270 360-90-180-270-360

Suppose we know that cos(60) = 0.5. By thinking about symmetry in the graph, how could we work out:

cos(120) = -0.5 cos(-60) = 0.5 cos(240) = -0.5 ? ? ?

Tan GraphWhat does it look like?

90 180 270 360-90-180-270-360 ?

Tan GraphWhat does it look like?

90 180 270 360-90-180-270-360

Suppose we know that tan(30) = 1/√3. By thinking about symmetry in the graph, how could we work out:

tan(-30) = -1/√3 tan(150) = -1/√3 ? ?

Laws of Trigonometric Functions

We saw for example sin(30) = sin(150) and cos(30) = cos(330). It’s also easy to see by looking at the graphs that cos(40) = sin(50). What laws does this give us?

sin(x) = sin(180 – x)

cos(x) = cos(360 – x)

sin and cos repeat every 360

tan repeats every 180

Bro Tip: These 5 things are pretty much the only thing you need to learn from this Chapter!

sin(x) = cos(90 – x)

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PracticeFind all the values in the range 0 to 360 for which sin/cos/tan will be the same.

sin(30) = sin(150)

cos(30) = cos(330)

sin(-10) = sin(190) = sin(350)

cos(-40) = cos(40) = cos (320)

sin(20) = cos(70)

sin(80) = sin(100)

cos(70) = cos(290)

cos(-25) = cos(25) = cos(335)

cos(80) = sin(10)

sin(15) = sin(165)

sin(-60) = sin(240) = sin(300)

tan(80) = sin(260)

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Dr Frost’s technique for remembering trig values(once described by a KGS tutee of mine as ‘the Holy Grail of teaching’)

0 45 90 30 60

sin 0 _1_√21 _1_

2√32

cos 1 _1_√20 √3

2_1_

2

tan 0 1 _1_√3√3

I literally picture this table in my head when I’m trying to remember my values.

All the surds in this block are √3All the surds in this block are √2

All the values in this square are over 2.

The diagonals starting from the top left are rational. The other values in the square are not.

I remember that out of tan(30) and tan(60), one is 1/√3 and the other √3. However, by considering the graph of tan, clearly tan(30) < tan(60), so tan(30) must be the smaller one, 1/√3

Practice

0 45 90 30 60

sin 0 _1_√21 _1_

2√32

cos 1 _1_√20 √3

2_1_

2

tan 0 1 _1_√3√3

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‘Magic Triangles’

You can easily work out sin(45), cos(45), sin(30), tan(30) etc. if you were ever to forget.

45

1

1√2

30

60

1

2 √3

sin(45) = _1_√2cos(30) = _√3_

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Angle quadrants

Given that sin α = 2/5, and that α is obtuse, find (without a calculator) the exact value of cos α.

ф

25

√21

Imagine working instead with the acute angle ф such that sin ф = 2/5

cos𝜙=√215

Therefore thinking about the graph of cos:

cos𝛼=− √215

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Since by thinking about the graph of cos, we can see that

Angle quadrants

Given that tan α = 5/12, and that α is acute, find the exact value of sin α and cos α.

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sin α = 5/13, cos α = 12/13

Given that cos α = -3/5, and that α is obtuse, find the exact value of sin α and tan α.

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sin α = 4/5, tan α = -4/3

Given that tan α = -√3, and that α is reflex, find the exact value of sin α and tan α.

3

sin α = -√3/2, cos α = 1/2

Hint: if tan α is negative, then is our reflex angle between 180 and 270, or 270 and 360?

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Onwards to Chapter 10...

The only 2 identities you need this chapter...

r

x = r cos

y = r sin

sin = y/r and cos = x/r and tan = y/x

1 sin cos

= tan

2 Pythagoras gives you... sin2 + cos2 = 1

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Examples of use

Simplify sin2 3 + cos2 3 = 1

Simplify 5 – 5sin2 = 5cos2

Show that:1 2

3This box is

intentionally left blank.

Given that p = 3 cos and q = 2 sin , show that 4p2 + 9q2 = 36.

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cos4𝜃− sin4 𝜃cos2𝜃

≡1− tan 2𝜃

Solving Trigonometric EquationsEdexcel May 2013 ()

Bro Tips for solving:1. If 0 ≤ < 180, then what range does 2 – 30 have?2. Immediately after the point at which you do sin-1 of both sides, list out the

other possible angles in the above adjusted range. Recall that sin(x) = sin(180-x) and that sin repeats every 360.

= 123.44, 176.57?

Solving Trigonometric EquationsEdexcel June 2010

tan = 0.4

tan 2x = 0.4 0 ≤ 2x < 7202x = 21.801, 201.801, 381.801, 561.801,x = 10.9, 100.9, 190.9, 280.9

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Solving Trigonometric EquationsEdexcel Jan 2010

Bro Tip: In general, when you have sin and cos, and one is squared, change the squared term to be consistent with the other.

(2sin x – 1)(sin x + 3) = 0sin x = 0.5 or sin x = -3

x = 30°, 150° ?

Edexcel Jun 2009

ExercisesEdexcel Jan 2009

Edexcel Jun 2008Edexcel Jan 2008

Edexcel Jan 2013

𝜃=−45 ° ,135 ° ,23.6 ° ,156.4 ° 𝑥=41.4 ° ,318.6 ° 284.5, 435.5, 644.5

65, 155

40 80 160 200 280 320

θ = 230.785, 309.23152, 50.8, 129.2

41.2, 85.5, 161.2

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Things to remember

If you square root both sides, don’t forget the . You’ll probably lose 2 marks otherwise.

Don’t forget solutions. If you have sin, you’ll always be able to get an extra solution by using 180 – x. If you have cos you can get an extra one using 360-x.

Remember that tan repeats every 180, sin/cos every 360.

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4 If you had sin2x and cos x, you’d replace the sin2 x with 1 – cos2 x. You’d then have a quadratic in terms of cos x which you can factorise.

5 Check whether the question expects you to give your answers in degrees or radians. If they say , then clearly they want radians.