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Reference Sheet: Programme F.9 — Trigonometry

Rules, procedures and drill, in the order of the book's programme. Every rule links to the unit in the lesson that derives it — if a rule here looks arbitrary, the link is where it stops being arbitrary.

Scope, as in the book: angles of right-angled triangles, so between 0^\circ and 90^\circ. General angles come in Programme F.11.

1. Angles as rotation

From Unit 01.

An angle is the amount a line turns about a point.

Name Size
full angle 360^\circ
straight angle 180^\circ
right angle 90^\circ
acute less than 90^\circ
obtuse between 90^\circ and 180^\circ

2. Degrees, minutes and seconds to decimal degrees

From Unit 01.

1^\circ=60' (minutes), 1'=60'' (seconds): base-60 place value.

D^\circ M'S''=D+\frac M{60}+\frac S{3600}\ \text{degrees}.

Examples. 45^\circ36'18''=45+0.6+0.005=45.605^\circ. 53^\circ29'7''=53+0.48333+0.00194=53.485^\circ (3 dp).

3. Decimal degrees to degrees, minutes and seconds

From Unit 01.

Multiply the fractional part of the degrees by 60 for minutes; multiply the fractional part of the minutes by 60 for seconds.

Examples. 18.478^\circ: 0.478\times60=28.68', 0.68\times60=40.8'', so 18^\circ28'41''. 236.986^\circ: 0.986\times60=59.16', 0.16\times60=9.6'', so 236^\circ59'10''.

4. Radians

From Unit 01.

A line of length r turning about one end has turned through 1 radian when its tip has travelled an arc of length r. A full turn is an arc of 2\pi r, so

360^\circ=2\pi\text{ rad},\qquad s=r\theta\ \ (\theta\text{ in radians}).

No unit means radians: \sin2 is the sine of 2 rad.

5. Converting between degrees and radians

From Unit 01.

\text{rad}=\text{deg}\times\frac\pi{180},\qquad\text{deg}=\text{rad}\times\frac{180}\pi;\qquad1^\circ=0.0175\text{ rad},\quad1\text{ rad}=57.296^\circ.

Leave common angles as multiples of \pi by writing the angle as a fraction of 180^\circ:

Degrees 30^\circ 45^\circ 60^\circ 90^\circ 120^\circ 150^\circ 180^\circ 270^\circ 360^\circ
Radians \frac\pi6 \frac\pi4 \frac\pi3 \frac\pi2 \frac{2\pi}3 \frac{5\pi}6 \pi \frac{3\pi}2 2\pi

Examples. 63.21^\circ=63.21\times\frac\pi{180}=1.1032 rad. 2.34\text{ rad}=134.1^\circ. \frac{7\pi}4=\frac74\times180^\circ=315^\circ.

6. Similar triangles

From Unit 02.

Triangles with the same three angles are similar: the same shape, possibly different sizes. Corresponding sides are in one common ratio,

\frac{AB}{A'B'}=\frac{AC}{A'C'}=\frac{BC}{B'C'},

so the ratio of two sides within a triangle is the same in every similar triangle. Sizes are fixed by the sides; the shape by the ratios.

Example. AB=2, AC=5, BC=4 cm, similar triangle with A'B'=3 cm: scale factor \frac32, so A'C'=7.5 cm and B'C'=6 cm.

7. The trigonometric ratios

From Unit 02.

In a right-angled triangle, relative to the acute angle \theta: the hypotenuse faces the right angle, the opposite side faces \theta, the adjacent side is the other side touching \theta.

\sin\theta=\frac{\text{opp}}{\text{hyp}},\qquad\cos\theta=\frac{\text{adj}}{\text{hyp}},\qquad\tan\theta=\frac{\text{opp}}{\text{adj}}=\frac{\sin\theta}{\cos\theta}.

8. Calculator values: degree and radian mode

From Unit 01 and Unit 02.

Set the mode to match the angle's unit before pressing \sin, \cos or \tan.

Examples. Degree mode: \sin27^\circ=0.4540, \cos84^\circ=0.1045, \tan43^\circ=0.9325. Radian mode: \cos1.321=0.2472, \tan0.013=0.0130, \sin\frac\pi6=0.5000. The same key press gives \sin2=0.9093 in radian mode and \sin2^\circ=0.0349 in degree mode.

9. Solving right-angled triangles

From Unit 02.

  1. Sketch the triangle; mark the angle and the known side.
  2. Name the known and unknown sides as opp, adj or hyp relative to that angle.
  3. Pick the ratio containing exactly those two sides, then rearrange.

Examples. A 3 m ladder at 56^\circ to the ground reaches 3\sin56^\circ=2.49 m up the wall. A ladder at 60^\circ whose top is 4.5 m up has length \frac{4.5}{\sin60^\circ}=5.20 m. A 50 m line at 12^\circ rises 50\sin12^\circ=10.40 m over a run of 50\cos12^\circ=48.91 m.

10. Reciprocal ratios

From Unit 02.

\operatorname{cosec}\theta=\frac1{\sin\theta}=\frac{\text{hyp}}{\text{opp}},\qquad\sec\theta=\frac1{\cos\theta}=\frac{\text{hyp}}{\text{adj}},\qquad\cot\theta=\frac1{\tan\theta}=\frac{\cos\theta}{\sin\theta}=\frac{\text{adj}}{\text{opp}}.

(Cosec is written \csc in some texts.) Evaluate with the reciprocal key. Useful when the hypotenuse is the unknown.

Examples. \cot12^\circ=4.7046, \sec37^\circ=1.2521, \operatorname{cosec}71^\circ=1.0576. A strut reaching 5 m up a wall at 43^\circ to the ground: L=5\operatorname{cosec}43^\circ=5\times1.4663=7.33 m.

11. Pythagoras' theorem

From Unit 03.

The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides: a^2+b^2=c^2 (c the hypotenuse). Proved by rearranging four copies of the triangle inside a square of side a+b.

Examples. Hypotenuse 8, one side 3: other side \sqrt{64-9}=\sqrt{55}=7.416. Side 5.6, hypotenuse 12.3: other side \sqrt{151.29-31.36}=\sqrt{119.93}=11.0 (1 dp). A rafter spanning 4.8 m and rising 1.4 m: \sqrt{23.04+1.96}=5.0 m.

12. The converse: a test for a right angle

From Unit 03.

If the squares on the two shorter sides add to the square on the longest, the triangle is right-angled (the right angle is opposite the longest side). If not, it is not.

Examples. 7,24,25: 49+576=625=25^2, right-angled. 5,11,12: 25+121=146\ne144, not right-angled.

13. The right-angled isosceles triangle

From Unit 04.

Half a square: angles 45^\circ,45^\circ,90^\circ, sides 1:1:\sqrt2.

\sin45^\circ=\cos45^\circ=\frac1{\sqrt2},\qquad\tan45^\circ=1\qquad(\text{the same for }\frac\pi4\text{ rad}).

Example. A prop at 45^\circ with its foot 3.4 m from the wall: L=3.4\sqrt2=4.81 m.

14. The half equilateral triangle

From Unit 04.

An equilateral triangle of side 2 cut by its altitude: angles 30^\circ,60^\circ,90^\circ, sides 1:\sqrt3:2.

\theta \sin\theta \cos\theta \tan\theta
30^\circ=\frac\pi6 \frac12 \frac{\sqrt3}2 \frac1{\sqrt3}
60^\circ=\frac\pi3 \frac{\sqrt3}2 \frac12 \sqrt3

Examples. A tree whose shadow is 8\sqrt3 m long, seen at 60^\circ elevation from the shadow's tip: height 8\sqrt3\tan60^\circ=24 m. A tent with an equilateral front and a \sqrt3 m pole: sides \frac{\sqrt3}{\sin60^\circ}=2 m.

15. The fundamental identity

From Unit 03.

Divide a^2+b^2=c^2 by c^2:

\cos^2\theta+\sin^2\theta\equiv1.

\equiv marks an identity, true for every angle. The ratio form is also a right-angle test.

Examples. 3,4,5: \frac9{25}+\frac{16}{25}=1, right-angled. 8,12,10: \left(\frac8{12}\right)^2+\left(\frac{10}{12}\right)^2=\frac{164}{144}\ne1, not right-angled.

16. Two more identities

From Unit 03.

Divide the fundamental identity by \cos^2\theta, then by \sin^2\theta:

1+\tan^2\theta\equiv\sec^2\theta,\qquad\cot^2\theta+1\equiv\operatorname{cosec}^2\theta.

17. Verifying identities

From Unit 05.

  1. Start from one side (usually the more complicated) and transform it into the other.
  2. Rewrite everything in \sin and \cos.
  3. Combine fractions over a common denominator.
  4. Replace 1-\cos^2\theta, 1-\sin^2\theta or \sin^2\theta+\cos^2\theta using §15.

Doing the same thing to both sides first is allowed only if it can be undone (multiplying by a non-zero quantity, yes; squaring, no). A numerical check can refute a claimed identity but never prove it.

Examples. \frac1{1-\cos\theta}+\frac1{1+\cos\theta}=\frac2{1-\cos^2\theta}=\frac2{\sin^2\theta}=2\operatorname{cosec}^2\theta. \tan\theta+\cot\theta=\frac{\sin^2\theta+\cos^2\theta}{\cos\theta\sin\theta}=\sec\theta\operatorname{cosec}\theta. \tan^2\theta-\sin^2\theta=\sin^2\theta(\sec^2\theta-1)=\sin^2\theta\tan^2\theta=\sin^4\theta\sec^2\theta.

18. Compound angles: cosine of a sum and a difference

From Unit 06.

\cos(\theta+\varphi)\equiv\cos\theta\cos\varphi-\sin\theta\sin\varphi,\qquad\cos(\theta-\varphi)\equiv\cos\theta\cos\varphi+\sin\theta\sin\varphi.

Proved from two stacked right-angled triangles (the sum), then by solving the sum formulas for the difference.

Examples. \cos75^\circ=\cos(45^\circ+30^\circ)=\frac1{\sqrt2}\cdot\frac{\sqrt3}2-\frac1{\sqrt2}\cdot\frac12=\frac{\sqrt3-1}{2\sqrt2}=0.2588. \cos15^\circ=\cos(60^\circ-45^\circ)=\frac{1+\sqrt3}{2\sqrt2}=0.9659.

19. Sums and differences of angles: sine and tangent

From Unit 06.

\sin(\theta\pm\varphi)\equiv\sin\theta\cos\varphi\pm\cos\theta\sin\varphi,
\tan(\theta+\varphi)\equiv\frac{\tan\theta+\tan\varphi}{1-\tan\theta\tan\varphi},\qquad\tan(\theta-\varphi)\equiv\frac{\tan\theta-\tan\varphi}{1+\tan\theta\tan\varphi}.

The tangent forms come from dividing the sine formula by the cosine formula, then top and bottom by \cos\theta\cos\varphi.

Example. Two slopes with \tan\theta=\frac12 and \tan\varphi=\frac13 combine to \tan(\theta+\varphi)=\frac{\frac12+\frac13}{1-\frac16}=\frac{5/6}{5/6}=1, so \theta+\varphi=45^\circ exactly (26.565^\circ+18.435^\circ).

20. Double angles

From Unit 06.

Put \varphi=\theta in the sum formulas:

\sin2\theta\equiv2\sin\theta\cos\theta,\qquad\cos2\theta\equiv\cos^2\theta-\sin^2\theta\equiv2\cos^2\theta-1\equiv1-2\sin^2\theta,\qquad\tan2\theta\equiv\frac{2\tan\theta}{1-\tan^2\theta}.

Example. \sin60^\circ=2\sin30^\circ\cos30^\circ=2\cdot\frac12\cdot\frac{\sqrt3}2=\frac{\sqrt3}2.

21. Sums and differences of ratios

From Unit 06.

\sin\theta+\sin\varphi\equiv2\sin\frac{\theta+\varphi}2\cos\frac{\theta-\varphi}2,\qquad\sin\theta-\sin\varphi\equiv2\cos\frac{\theta+\varphi}2\sin\frac{\theta-\varphi}2,
\cos\theta+\cos\varphi\equiv2\cos\frac{\theta+\varphi}2\cos\frac{\theta-\varphi}2,\qquad\cos\theta-\cos\varphi\equiv-2\sin\frac{\theta+\varphi}2\sin\frac{\theta-\varphi}2.

(Sum-to-product.) Note the -2 in the last. Each is a product formula (§22) read backwards with A=\frac{\theta+\varphi}2, B=\frac{\theta-\varphi}2.

Example. \sin5\theta+\sin3\theta\equiv2\sin4\theta\cos\theta.

22. Products of ratios

From Unit 06.

2\sin\theta\cos\varphi\equiv\sin(\theta+\varphi)+\sin(\theta-\varphi),
2\cos\theta\cos\varphi\equiv\cos(\theta+\varphi)+\cos(\theta-\varphi),
2\sin\theta\sin\varphi\equiv\cos(\theta-\varphi)-\cos(\theta+\varphi).

(Product-to-sum.) Add or subtract the sum and difference formulas; the cross terms cancel.

Example. 2\sin45^\circ\cos15^\circ=\sin60^\circ+\sin30^\circ=\frac{\sqrt3+1}2=1.3660.

Traps

  • Degrees in s=r\theta. The formula needs radians: a 0.2 m pulley turning 90^\circ passes 0.2\times\frac\pi2=0.314 m of belt, not 0.2\times90=18 m.
  • The wrong calculator mode. \sin2 is 0.9093 (radians), not 0.0349 (degrees). No unit means radians.
  • DMS as a decimal. 18^\circ28' is 18.467^\circ, not 18.28^\circ: minutes are sixtieths, not hundredths.
  • Ratios proportional to the angle. \sin24^\circ is not 2\sin12^\circ, and \sin60^\circ=0.866, not 2\sin30^\circ=1.
  • Naming sides from the wrong angle. Opposite and adjacent swap when the other acute angle is used; the hypotenuse never changes.
  • Adding sides instead of squares. A 4.8 m by 1.4 m rafter is \sqrt{4.8^2+1.4^2}=5.0 m, not 6.2 m.
  • Testing the converse against the wrong side. Compare the sum of the two shorter squares with the longest: for 5,11,12, 25+121\ne144.
  • \sin^{-1} or \operatorname{cosec} confusion. \operatorname{cosec}\theta=\frac1{\sin\theta}; \sin^{-1} is the inverse function (F.11), a different thing.
  • Proving an identity by a spot check. 2\sin\theta\cos\theta and \tan\theta agree at 45^\circ but not at 30^\circ (0.866 vs 0.577). A check refutes; only a chain of identities proves.
  • Sharing a ratio over a sum. \cos(45^\circ+30^\circ)\ne\cos45^\circ+\cos30^\circ=1.573, which exceeds 1; the true value is 0.2588.

Self-check

  1. Convert 27^\circ41'15'' to decimal degrees.
  2. Convert 104.372^\circ to degrees, minutes and seconds.
  3. Express 210^\circ and 22.5^\circ as multiples of \pi, and convert 1.2 rad to degrees (1 dp).
  4. A pulley of radius 0.25 m turns through 150^\circ. How much belt passes over it?
  5. Find, to 4 dp, \sin38^\circ, \cos1.1 and \cot25^\circ.
  6. A 6.5 m ladder makes 72^\circ with the ground. How high up the wall does it reach, and how far out is its foot?
  7. A guy wire is fixed 12 m up a mast and meets the ground at 55^\circ. Use cosec to find its length.
  8. A right-angled triangle has hypotenuse 13.5 and one side 6.2. Find the third side to 2 dp.
  9. Is a triangle with sides 20, 21, 29 right-angled? One with sides 6, 9, 11?
  10. Using the special triangles, find \tan30^\circ+\tan60^\circ exactly.
  11. Verify \sec\theta-\cos\theta\equiv\sin\theta\tan\theta.
  12. (a) Use a double-angle formula to find \sin15^\circ\cos15^\circ exactly. (b) Write \cos7\theta+\cos3\theta as a product.

Fully worked solutions

1. 27+\frac{41}{60}+\frac{15}{3600}=27+0.68333+0.00417=27.6875^\circ.

2. 0.372\times60=22.32'; 0.32\times60=19.2''. So 104.372^\circ=104^\circ22'19''.

3. 210^\circ=\frac{210}{180}\pi=\frac{7\pi}6; 22.5^\circ=\frac{22.5}{180}\pi=\frac\pi8; 1.2\times\frac{180}\pi=68.755^\circ=68.8^\circ.

4. 150^\circ=\frac{150}{180}\pi=\frac{5\pi}6=2.6180 rad, so s=r\theta=0.25\times2.6180=0.6545 m.

5. Degree mode: \sin38^\circ=0.6157. Radian mode: \cos1.1=0.4536. Degree mode: \tan25^\circ=0.46631, so \cot25^\circ=\frac1{0.46631}=2.1445.

6. The ladder is the hypotenuse. Height =6.5\sin72^\circ=6.5\times0.95106=6.18 m; foot =6.5\cos72^\circ=6.5\times0.30902=2.01 m.

7. The wire is the hypotenuse and 12 m is opposite the 55^\circ angle: L=12\operatorname{cosec}55^\circ=12\times\frac1{0.81915}=12\times1.22077=14.65 m.

8. \sqrt{13.5^2-6.2^2}=\sqrt{182.25-38.44}=\sqrt{143.81}=11.99.

9. 20^2+21^2=400+441=841=29^2: right-angled. 6^2+9^2=36+81=117\ne121=11^2: not right-angled.

10. From the half equilateral triangle, \tan30^\circ=\frac1{\sqrt3} and \tan60^\circ=\sqrt3: \frac1{\sqrt3}+\sqrt3=\frac{1+3}{\sqrt3}=\frac4{\sqrt3}=\frac{4\sqrt3}3=2.3094.

11. \text{LHS}=\frac1{\cos\theta}-\cos\theta=\frac{1-\cos^2\theta}{\cos\theta}=\frac{\sin^2\theta}{\cos\theta}=\sin\theta\cdot\frac{\sin\theta}{\cos\theta}=\sin\theta\tan\theta=\text{RHS}.

12. (a) \sin2\theta\equiv2\sin\theta\cos\theta with \theta=15^\circ: \sin15^\circ\cos15^\circ=\frac12\sin30^\circ=\frac12\cdot\frac12=\frac14. (b) \cos\theta+\cos\varphi\equiv2\cos\frac{\theta+\varphi}2\cos\frac{\theta-\varphi}2 with 7\theta and 3\theta: \cos7\theta+\cos3\theta\equiv2\cos5\theta\cos2\theta.