C1: Chapters 8 & 10 Trigonometry
Dr J Frost ([email protected])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_
2
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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.
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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.
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