Grade 11 Trigonometry: Short Notes

11 ශ්‍රේණිය · 18 පරිච්ඡේදය · ත්‍රිකෝණමිතිය

43%

of Grade 11 term tests include it (26 of 61)

280

marks in past papers (40 questions)

12 min

full lesson with quick checks

🎯 What you'll learn

Name the hypotenuse, opposite and adjacent sides for a given angle
Write sin, cos and tan of an angle in a right triangle, and use the values for 30°, 45° and 60°
Find an unknown side or angle using trigonometric tables
Solve problems with angles of elevation and depression
Use trigonometry with bearings to find distances and directions

How tall is that coconut tree?

Your uncle wants to know if a coconut tree could hit the roof if it falls. Nobody wants to climb it with a tape. Stand 20 m away, measure one angle, and a little maths gives the height. That's trigonometry: finding lengths and angles you can't measure directly.

Name the sides from the angle

For the angle θ at A:• Hypotenuse AC (amber): opposite the right angle, the longest side.• Opposite side BC (red): across from θ.• Adjacent side AB (blue): next to θ, not the hypotenuse.If you use the angle at C instead, opposite and adjacent swap.

🧮 The three ratios

sin = opposite ÷ hypotenusecos = adjacent ÷ hypotenusetan = opposite ÷ adjacentA memory trick: SOH CAH TOA. They work only in a right-angled triangle.

sin θ = BC/AC cos θ = AB/AC tan θ = BC/AB

✏️ From one ratio to the others

θ is an acute angle and sin θ = 5/13. Find cos θ and tan θ without tables.

1Sketch a right triangle: opposite = 5, hypotenuse = 13.
2Adjacent² = 13² − 5² = 169 − 25 = 144, so adjacent = 12.
3cos θ = 12/13 and tan θ = 5/12.

✅ cos θ = 12/13, tan θ = 5/12

Special angles: 30°, 45°, 60°

Read these straight off the two triangles:• sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3• sin 45° = cos 45° = 1/√2, tan 45° = 1• sin 60° = √3/2, cos 60° = ½, tan 60° = √3

Using the trig tables

For other angles, read the value from the tables (4 decimal places).• Finding a side: write the ratio with x, then rearrange. E.g. x/20 = tan 35°, so x = 20 × 0.7002.• Finding an angle: work out the ratio as a decimal, then look in the body of the table for the nearest value. E.g. cos θ = 0.6 gives θ ≈ 53°.

✏️ Back to the coconut tree

From a point on level ground 20 m from the foot of the tree, the angle up to the top of the tree is 35°. Find the height of the tree to the nearest 0.1 m. (tan 35° = 0.7002)

1Height h is opposite 35°; the 20 m is adjacent. Use tan.
2tan 35° = h/20, so h = 20 × 0.7002.
3h = 14.004 ≈ 14.0 m.

✅ 14.0 m

⚠️ Choosing the wrong ratio

The most common mistake in recent O/L papers: using the wrong ratio, e.g. the opposite side for cos. Before writing any ratio:1. Mark the angle you are using.2. Label the three sides H, O and A from that angle.3. Pick the ratio that links the side you know with the one you want.

Elevation and depression

Both are measured from the horizontal. Looking up from P to the top T: angle of elevation. Looking down from T to P: angle of depression. They are alternate angles between parallel horizontals, so they are equal.

⚠️ Sketch first, mark everything

Examiners note year after year that students don't mark the given data on a sketch. Draw the horizontal line, put the angle between the horizontal and the line of sight, never against the vertical pole, and write each length on the diagram. A clear right triangle makes the right ratio obvious.

Bearings meet trigonometry

A bearing is measured clockwise from north, written with three figures: 060°, 135°, 270°.In exam questions, two directions are often at right angles (e.g. due north and due east). Sketch the north lines at each point, find the right triangle, then use sin, cos or tan.

📌 Remember

• Label sides from the angle: hypotenuse, opposite, adjacent.• SOH CAH TOA, in right triangles only.• Know sin, cos, tan of 30°, 45°, 60° by heart.• Elevation and depression are measured from the horizontal and are equal.• Bearings: clockwise from north, three figures. Always sketch and mark the data first.

🧾 Quick revision sheet

sin θ	opposite ÷ hypotenuse
cos θ	adjacent ÷ hypotenuse
tan θ	opposite ÷ adjacent
sin 30° and cos 60°	Both equal ½
tan 45°	1
tan 60°	√3 (about 1.732)
Angle of depression	Angle below the horizontal when looking down; equals the angle of elevation from below.
How is a bearing measured?	Clockwise from north, in three figures (e.g. 045°).