Notes · Unit 6
Trigonometry and Bearing
Thirteen short slides on angle measure, arc length and sector area, the unit circle, trigonometric ratios, identities, elevation and depression, and bearings.
Press Next (or the right arrow key) to move on. On most slides, each press reveals one more idea.
RadiansArc and sectorUnit circleIdentitiesElevationBearings
1 · Angle measure
Degrees and radians
Sexagesimal system. 1° = 60′ and 1′ = 60″, so 1° = 3600″. For example 60°30′ = 60 + 30/60 = 60.5°.
Direction. An anticlockwise turn from the initial side gives a positive angle. A clockwise turn gives a negative angle.
One radian is the central angle of an arc whose length equals the radius. A full turn is 360° = 2π radians.
Convert. π rad = 180°, so 1 rad = (180/π)° ≈ 57.2958°.
45° = 45 × π/180 = π/4 · 2π/5 = 2π/5 × 180/π = 72°
45° = 45 × π/180 = π/4 · 2π/5 = 2π/5 × 180/π = 72°
2 · Arc length and sector area
Radians make the formulas simple
Arc length
l = rθ
Sector area
A = ½r2θ
θ must be in radians, because a radian is the ratio of arc to radius. Put θ = 2π and you get l = 2πr, the circumference.
Arc. r = 6 cm, angle 120° = 2π/3 rad. l = 6 × 2π/3 = 4π cm
Sector. r = 12 cm, θ = π/3. A = ½ × 144 × π/3 = 24π cm2
For a fixed radius, the sector area is directly proportional to the angle. Double the angle and the area doubles.
3 · Angles in standard position
Quadrantal and co-terminal angles
Quadrantal angle. The terminal ray lies on an axis: 0°, 90°, 180°, 270°, 360° and so on, all multiples of 90°. So 45° is not quadrantal.
Co-terminal angles share the same terminal ray, so they differ by a whole number of full turns (360°).
390° − 30° = 360° and 60° − (−300°) = 360°.
390° − 30° = 360° and 60° − (−300°) = 360°.
Co-terminal angles have the same trigonometric ratios, because the ratios depend only on where the terminal ray meets the unit circle.
4 · The unit circle
Where sine and cosine come from
x2 + y2 = 1
The point at angle θ is P(cos θ, sin θ). So cos θ = x and sin θ = y.
A point on the unit circle is never more than 1 unit from the axes, so sin θ is never greater than 1.
tan θ = y/x. At 90° the point is (0, 1), so tan 90° = 1/0 is undefined.
cosec θ = 1/sin θ, so cosec 30° = 1/(1/2) = 2 and cosec 0° is undefined because sin 0° = 0.
5 · Values to know
Special angles
sin
sin 0° = 0sin 30° = 1/2sin 45° = 1/√2sin 60° = √3/2sin 90° = 1
cos
cos 0° = 1cos 30° = √3/2cos 45° = 1/√2cos 60° = 1/2cos 90° = 0
tan = sin ÷ cos
tan 30° = 1/√3tan 45° = 1tan 60° = √3
On the axes. cos 180° = −1 (point (−1, 0)) and sin 270° = −1 (point (0, −1)).
Example. sin 60° cos 30° − cos 60° sin 30° = 3/4 − 1/4 = 1/2
6 · Signs
Which ratio is positive where?
Quadrant I
All ratios positive.
Quadrant II
Only sin (and cosec) positive. y > 0 but x < 0.
Quadrant III
Only tan (and cot) positive. Both x and y are negative, so sin and cos are negative but y/x is positive.
Quadrant IV
Only cos (and sec) positive.
Example. If cos θ = −3/5 and θ is in quadrant II, then sin θ = 4/5 (positive). So tan θ = (4/5) ÷ (−3/5) = −4/3.
7 · Identities
True for every angle
sin2θ + cos2θ = 1
It comes from x2 + y2 = 1, so it holds for every angle, not only acute ones.
It comes from x2 + y2 = 1, so it holds for every angle, not only acute ones.
tan2θ + 1 = sec2θ
Divide the first identity by cos2θ.
Divide the first identity by cos2θ.
Not an identity: sin θ + cos θ = 1. At 30° the sum is 1/2 + √3/2 ≈ 1.37. It happens to equal 1 at 0° and 90°, but that does not make it an identity.
8 · Right-angled triangles
Elevation and depression
tan θ = perpendicular ÷ base. If one acute angle is θ, the other is 90° − θ.
Angle of elevation: measured from the horizontal up to the line of sight. The angle of depression is measured down from the horizontal. Horizontal lines are parallel, so the depression of a boat from a tower equals the elevation of the tower from the boat.
Tree. 20 m from its foot, elevation 45°. height = 20 × tan 45° = 20 m
Tower. 60 m high, car seen at depression 60°. tan 60° = 60/d, so d = 60/√3 = 20√3 m.
Kite. String 50 m at 30° to the ground. height = 50 × sin 30° = 25 m
9 · Bearings
Direction as an angle from north
A bearing is the angle measured clockwise from north, written with three digits: north 000°, east 090°, south 180°, west 270°.
North-west lies halfway between west (270°) and north (360°), so its bearing is 315°.
Back bearing. Seen from the other end, the line turns 180°. If the bearing of B from A is 040°, the bearing of A from B is 040° + 180° = 220°.
Walking. 5 km north, then 5 km east: tan θ = 5/5 = 1, so the end point is at bearing 045°.
A bearing of 120° lies between 090° (east) and 180° (south), so B is to the south-east of A.
10 · Common mistakes
Five traps to avoid
Degrees in the arc formula. l = rθ works only when θ is in radians.
Thinking the identity works only for acute angles. sin2θ + cos2θ = 1 holds for all angles.
Ignoring the quadrant. Check the sign of sin, cos and tan before you finish.
Writing tan 90° = ∞ as a number. It is undefined.
Bearings from the wrong direction. Measure clockwise from north, with three digits.
11 · Quick check
Try these, then tap to check
12 · Your turn
Practise Unit 6
Three timed MCQ tests with an explanation after every answer.
Practice questions are independently written and are not official FBISE papers.