Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

TRIGONOMETRY
Hints on solving trig problems
• If no diagram is given, draw one yourself
• Mark the right angles in the diagram
• Show the sizes of other angles and the lengths of any lines that are shown
• Mark the angles or sides you have to calculate
• Consider whether you need to create right triangles by drawing extra lines, e
...
divide an isosceles
triangle into 2 congruent right angles
• Decide whether you will use Pythagoras theorem, sine, cosine or tangent
• Check that your answer is reasonable
• The hypotenuse is the longest side in a right angled triangle
• Remember:
-opposite does not make a reff angle
-adjacent: a line that joins reff angle to 90°

Ø

e
...
Soh Cah Toa
√3𝑡𝑎𝑛Ø = 1
𝑡𝑎𝑛Ø =

1
√3

=

𝑂
𝐴

Finding this side we use Pythagoras
𝐻2= (√3) 2 +𝐶𝐷 2
= 3+1
√𝐻2 = √4 (remember that √ cancels 2)
𝐻=2
Therefore the correct diagram is as follows:

1

2
1

Ø
√3

What you need to know:
• Quadrants (including multiple rotations)
• Pythagoras theorem (to find the unknown sides)
• Reduction formulae
• Special angles
• Identities (square Identity and quotient Identity)
• How to find the general solutions for sin, cos and tan
• Compound angles and Double angles

-tan
-cos

All +tive

S
90° + Ø & 180° - Ø

A
90° - Ø

-cos

-sin

-sin

-tan

270° - Ø & 180° + Ø

T

270° + Ø & 360° - Ø

C

2

Quadrants
This is the second quadrant and it’s rotation( the opposite direction is quadrant 1)
M

N

y

0

x

This is the 4th quadrant and it’s rotation ( the opposite direction is quadrant 3 with a not full rotation)

y

0

N
...
g
...
𝐜𝐨𝐬𝟐𝟐𝟓°

tan(360° + 120°)
...
cos14°
...
cos(180° + 45°)

tan(180° − 60°)
...
(−sin45°)
−cos45°

(−tan60°)
...
(−sin45°)
−cos45°
√3
...


1
√2

(remember that √ cancels √ )

1
√2

=

3
2

e
...
2 𝒄𝒐𝒔𝟐𝟔𝟎°
...
𝒄𝒐𝒔𝟑𝟓𝟎°

𝑐𝑜𝑠(180° + Ø)
...
𝑠𝑖𝑛(180° + 10°)
...
(−𝑐𝑜𝑠10°)
𝑐𝑜𝑠80°
...
(𝑐𝑜𝑠10°)

1

= − 𝑠𝑖𝑛10 °

5

Special angles
Ø
SinØ

0°
0

CosØ

tanØ

30°

45°

60°

1
2

√2
2

√3
2

1

√3
2

√2
2

1
2

0

√1

1

√3

90°
1

0

Infinite

√3

Identities

90° − Ø

r
y

Ø
x
Soh Cah Toa
𝒔𝒊𝒏Ø =

𝒚
𝒓

𝑐𝑜𝑠(90° − Ø) =

𝑦
𝑟

𝑥

𝑐𝑜𝑠Ø = 𝑟

𝒚 𝟐

𝒙 𝟐

• 𝒔𝒊𝒏𝟐 Ø + 𝒄𝒐𝒔𝟐 Ø = ( 𝒓) + (𝒓)
𝑌2

𝑥2

= 𝑟 + 𝑟2

=𝑦 2 + 𝑥 2
𝑟²
𝑟2

=𝑟 2

6

=1
•𝒔𝒊𝒏𝟐 Ø = 𝟏 − 𝒄𝒐𝒔𝟐 Ø
=𝑐𝑜𝑠 2 Ø − 1 − 𝑠𝑖𝑛2 Ø
𝑠𝑖𝑛Ø = √1 − 𝑐𝑜𝑠 2 − Ø
𝑐𝑜𝑠Ø = √1 − 𝑠𝑖𝑛2 Ø

Quotient Identity
𝒚

𝒙

•𝒔𝒊𝒏Ø ÷ 𝒄𝒐𝒔Ø = 𝒓 ÷ 𝒓
𝑌


...
𝑐𝑜𝑠Ø = 𝑡𝑎𝑛Ø
Exercise
a
...
cosØ
2
...
sin²Ø+cos²Ø
Solution:
𝑥

1
...
2𝑡𝑎𝑛Ø + 3 = 2 (−
=

2
√3

1
−√3

)

+3
√3

3
...
Given cos16°=m
Express the following in terms of m:
1
...
2cos²74°-1
3
...
𝑠𝑖𝑛16° = √1 − 𝑚2
2

2
...
𝑡𝑎𝑛16° = √1 − 𝑚2
m
Practice questions for you:
Simplify without using a calculator:
1
...
𝑐𝑜𝑠330°
...


𝑐𝑜𝑠750°
...
cos(−Ø)
Cos(360° − Ø)
...
sin(180 − Ø)

3
...
𝑠𝑖𝑛300
...
sin(−135°)
𝑠𝑖𝑛104°
...


𝑐𝑜𝑠260°
...
𝑠𝑖𝑛190°

---