Trigonometry (II)
Geometry · Trigonometry (II)
Syllabus tag: KCSE | Mathematics | Form 3 | Topic 3 Trigonometry (II)
Lesson objectives
By the end of this topic, you should be able to:
- Define and draw the unit circle and use it to find trigonometric ratios.
- Find trigonometric ratios of angles greater than 90° and of negative angles.
- State and apply the sine rule and the cosine rule.
- Solve triangles that are not right-angled.
Trigonometry (II)
Form 2 trigonometry needed a right angle. This topic removes that restriction and handles angles beyond 90°.
a) The unit circle
Draw a circle of radius 1 centred at the origin.
Take a point P on the circle at angle θ, measured anticlockwise from the positive x-axis.
Then cos θ is the x-coordinate of P, and sin θ is the y-coordinate.
This definition works for any angle, not just those under 90°, which is exactly what was needed.
Since the radius is 1, Pythagoras gives sin²θ + cos²θ = 1 immediately. That identity is used constantly later.
b) Angles beyond 90°
The signs follow from the coordinates.
In the second quadrant x is negative and y positive, so only sine is positive.
In the third both are negative, so only tangent, being their ratio, is positive.
In the fourth x is positive and y negative, so only cosine is positive.
Remember it as ASTC, reading anticlockwise from the first quadrant.
c) Related angles
Any angle relates to an acute angle in the first quadrant.
sin(180° − θ) = sin θ. cos(180° − θ) = −cos θ.
sin(180° + θ) = −sin θ. cos(180° + θ) = −cos θ.
sin(360° − θ) = −sin θ. cos(360° − θ) = cos θ.
Find the acute angle first, read its ratio, then attach the sign from ASTC.
d) Negative angles
A negative angle is measured clockwise.
sin(−θ) = −sin θ, while cos(−θ) = cos θ.
e) The sine rule
a/sin A = b/sin B = c/sin C. Each side is opposite the angle of the same letter.
Use it when you have two angles and a side, or two sides and a non-included angle.
Turn it upside down when finding an angle: sin A / a = sin B / b.
The ambiguous case. Given two sides and a non-included angle, two triangles may fit. That is because sin θ and sin(180° − θ) are equal. Check the obtuse option before discarding it.
f) The cosine rule
a² = b² + c² − 2bc cos A. Here A is the angle between sides b and c.
Use it when you have three sides, or two sides and the included angle.
To find an angle from three sides, rearrange: cos A = (b² + c² − a²) / 2bc.
Notice that if A is 90°, cos A is 0 and the rule reduces to Pythagoras. The cosine rule is Pythagoras generalised to any triangle.
g) Choosing the rule
Two angles and a side, or two sides and a non-included angle: sine rule.
Three sides, or two sides and the included angle: cosine rule.
Right-angled: use SOH CAH TOA, which is quicker.
h) Where this is used
Surveying inaccessible distances. Navigation. Force diagrams. Any triangle problem without a right angle.
Words to know
- Unit circle -- a circle of radius 1 centred at the origin, used to define the ratios.
- Related acute angle -- the acute angle a given angle makes with the x-axis.
- Sine rule -- a ÷ sin A = b ÷ sin B = c ÷ sin C.
- Cosine rule -- a² = b² + c² − 2bc cos A.
- Ambiguous case -- where two triangles fit the same SSA data.
:::checkpoint Check yourself
- In which quadrant is only the tangent positive?
- Write sin 150° in terms of an acute angle.
- A triangle has b = 5, c = 8 and A = 60°. Find a.
- Why does the cosine rule reduce to Pythagoras when A is 90°? :::
Bridge to practice
The exercises begin with the unit circle and CAST, move through related acute angles and negative angles, and finish with the sine and cosine rules and their selection. For every non-right-angled triangle, list what you are given before choosing a rule.