Trigonometry Exam-Prep Notes

Introduction – What is Trigonometry?

• Study of relationships between the angles and side-lengths of triangles.
• Applications: navigation, surveying, engineering, architecture.
• Angles traditionally denoted by Greek letters: $\theta,\;\alpha,\;\beta,\;\phi.$

Right-Angled Triangles – Naming the Sides

• In a right-angled triangle the side opposite the right angle is the hypotenuse (longest side).
• The other sides are named opposite & adjacent with respect to a chosen reference angle.
• Example ΔABC (right-angled at B):
– Using $\hat A=\theta$ as reference →
▪ $AC$ (across from $\hat B$ ) = hypotenuse = “little b”.
▪ $AB$ (across from $\hat A$ ) = opposite = “little c”.
▪ $BC$ (next to $\hat A$ ) = adjacent = “little a”.
• Pythagoras in right-angled triangles: $a^{2}+b^{2}=c^{2}.$

Fundamental Trigonometric Ratios (SOH-CAH-TOA)

• $\displaystyle \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$
• $\displaystyle \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$
• $\displaystyle \tan\theta=\frac{\text{opposite}}{\text{adjacent}}$
• Lazy mathematician shorthand:
– “Determine $\sin C$ ” ≡ “ratio (opp/hyp) from angle C”.
– “Find $\cos C$ ” ≡ “ratio (adj/hyp) from angle C”.
– “Give $\tan C$ ” ≡ “ratio (opp/adj) from angle C”.

Worked Example 1 (ΔDEF)

• Given right-angled ΔDEF with $DE=5\text{ cm},\;EF=12\text{ cm},\;\hat E=90^\circ$ .
a) Hypotenuse $DF$ : $DF=\sqrt{5^{2}+12^{2}}=13\text{ cm}$ .
b) With $\hat F=\theta$ (same as $\beta$ ):
▪ $\sin\theta=\dfrac{5}{13}$ , $\cos\theta=\dfrac{12}{13}$ , $\tan\theta=\dfrac{5}{12}$ .
c) With $\hat D=\beta$ :
▪ $\sin\beta=\dfrac{12}{13}$ , $\cos\beta=\dfrac{5}{13}$ , $\tan\beta=\dfrac{12}{5}$ .

Calculator Skills – Ratio ➜ Number (Deg-Mode!)

• Ensure calculator is in degree mode.
• Examples:
– $2\sin 30^\circ=2(0.5)=1.0$ .
– $(2\sin 30^\circ)^{2}=1^{2}=1$ .
– $2\sin 42^\circ\cos42^\circ\approx2(0.6691)(0.7431)=0.993$ .

Calculator Skills – Number ➜ Angle (INV keys)

• Use $\sin^{-1},\;\cos^{-1},\;\tan^{-1}$ (SHIFT keys).
• Example set:
– If $\cos\theta=0.5$ → $\theta=\cos^{-1}(0.5)=60.0^\circ$ .
– If $\tan\theta=4.123$ → $\theta=\tan^{-1}(4.123)\approx76.3^\circ$ .
– Solve $2\sin\theta=1.124$ : $\sin\theta=0.562\;\Rightarrow\;\theta=34.2^\circ.$ (all acute solutions inside $0^\circ\le\theta\le90^\circ$ ).

Finding a Missing Side in a Right-Angled Triangle

• Choose ratio containing the known angle & one known side.
• Example ΔFUN: $FN=10,\;\hat U=90^\circ,\;\hat N=28^\circ$ , need $UF$ .
– Adjacent (FN) & hypotenuse (UN) known? No. Instead use $\cos28^\circ=\dfrac{\text{adjacent}(FN)}{\text{hypotenuse}(UN)}$ etc.
– Full working leads to $UF\approx4.7$ (rounded one decimal).

Finding a Missing Angle from Two Sides

• Apply inverse functions.
• Example: ΔPQR with $PQ=47,\;QR=60,\;PR=40$ , find $\hat Q$ .
– $\cos\hat Q=\dfrac{\text{adjacent}=47}{\text{hypotenuse}=60}\approx0.7833$ → $\hat Q\approx38.6^\circ$ .

Quick Self-Check Questions (booklet pg 10–11)

• Express $\tan\theta,\;\sin\theta,\;\cos\theta$ in terms of sides in ΔABC (right-angled at B).
– $\tan\theta=\dfrac{AB}{BC},\;\sin\theta=\dfrac{AB}{AC},\;\cos\theta=\dfrac{BC}{AC}$ .
• Evaluate combinations e.g. $\sin(\alpha+\beta),\;\sin\alpha\cos\beta,\;\tan^{2}\beta$ with given $\alpha=48^\circ,\;\beta=26^\circ$ etc. (calculator practice).
• Solve trig equations within $0^\circ\le x\le90^\circ$ :
– $7\cos3x=\tan^{2}x$ , $\cos58.3^\circ+\tan88.5^\circ=\sin2x$ , etc.

Angles of Elevation & Depression

• Defined in the vertical plane measured from the horizontal.
• Key fact: angle of depression from A to C equals angle of elevation from C to A.
• Tree example: boat 20 m from bank, elevation $20^\circ$ ⇒ tree height $=20\tan20^\circ\approx7.3\text{ m}$ .
• Cliff example: depression $55^\circ$ , horizontal distance $70\text{ m}$ :
– Elevation from boat is also $55^\circ$ .
– Height $h=70\tan55^\circ\approx100.1\text{ m}$ .
• London Olympic sculpture problem: distance 100 m, elevation $30.9^\circ$ , depression $28.8^\circ$ . Use two-right-triangle setup → height $BD\approx101\text{ m}$ (nearest m).

Trigonometry in the Cartesian Plane (0°–360°)

• A point $P(x,y)$ on a circle of radius $r$ centered at origin gives identities:
– x^{2}+y^{2}=r^{2}\;(r>0).
– $\sin\theta=\dfrac{y}{r},\;\cos\theta=\dfrac{x}{r},\;\tan\theta=\dfrac{y}{x}$ (when $x\neq0$ ).
• Helpful rhyme: “S C T – Y X Y over R R X” (maps $\sin\to y/r$ etc.).

Quadrants & CAST Diagram

• Quadrant I: 0^\circ<\theta<90^\circ – all ratios $+$.
• Quadrant II: 90^\circ<\theta<180^\circ – only $\sin$ positive.
• Quadrant III: 180^\circ<\theta<270^\circ – only $\tan$ positive.
• Quadrant IV: 270^\circ<\theta<360^\circ – only $\cos$ positive.
• CAST mnemonic starting from fourth quadrant clockwise:
– C (cos), A (all), S (sin), T (tan).

Ratio-&-Restriction Method (No Calculator)

Step-by-step (example given $3\tan\theta/4=1$ with $\theta\in[180^\circ;360^\circ]$ ):

Isolate: $\tan\theta=-\tfrac43$ .
Sign & quadrant: \tan<0 in QII & QIV. Restriction ⇒ QIV.
Sketch a triangle inside QIV: take $\text{opp}=4,\;\text{adj}=-3$ → $r=5$ .
Compute required expression e.g. $25\sin^{2}\theta-5\cos\theta$ by substitutions $\sin=-\frac{4}{5},\cos=\frac{3}{5}$ .

Special Angles Table (must be known)

Angle $\theta$	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
$\sin\theta$	$0$	$\tfrac12$	$\tfrac{\sqrt2}{2}$	$\tfrac{\sqrt3}{2}$	$1$	$0$	$-1$	$0$
$\cos\theta$	$1$	$\tfrac{\sqrt3}{2}$	$\tfrac{\sqrt2}{2}$	$\tfrac12$	$0$	$-1$	$0$	$1$
$\tan\theta$	$0$	$\tfrac{1}{\sqrt3}$	$1$	$\sqrt3$	undefined	$0$	undefined	$0$

Classic No-Calculator Evaluations

• $3\sin30^\circ\tan45^\circ\cos30^\circ=3(\tfrac12)(1)(\tfrac{\sqrt3}{2})=\tfrac{3\sqrt3}{4}$ .
• $4\sin60^\circ\tan45^\circ+2\cos30^\circ=4(\tfrac{\sqrt3}{2})(1)+2(\tfrac{\sqrt3}{2})=3\sqrt3$ .
• $\dfrac{4\cos45^\circ-3\sin30^\circ}{2}=\dfrac{4(\tfrac{\sqrt2}{2})-3(\tfrac12)}{2}=\dfrac{2\sqrt2-1.5}{2}$ .
• $2\tan60^\circ\cos0^\circ-\sin90^\circ=2(\sqrt3)(1)-1=2\sqrt3-1$ .
• $1-\dfrac{1}{2}\sin30^\circ=1-\dfrac{1}{2}(\tfrac12)=\dfrac{3}{4}.$

Revision Worksheet Highlights

• Identify trig ratio from side labels (a,b,c).
• ΔPQR with $PQ=4, an\hat P=\tfrac32\Rightarrow QR=6,PR=\sqrt{52}=2\sqrt{13}$ , $\hat Q=56.3^\circ$ .
• Compound-angle numeric substitutions e.g. $\cos(2y-x)$ .
• Solve equations such as $3\cos3\theta=1.056$ inside $0^\circ-90^\circ$ .
• Cartesian: point (4,-5):
– $r=\sqrt{4^{2}+(-5)^{2}}=\sqrt{41}$ .
– $\sin\alpha=-\dfrac{5}{\sqrt{41}},\;\cos\alpha=\dfrac{4}{\sqrt{41}},\;\tan\alpha=-\dfrac{5}{4}$ .
– Evaluate expressions like $-3\cos\alpha+(\sin^{2}\alpha+\cos^{2}\alpha)$ .

Real-World Applications

• Leaning Tower of Pisa: given true height 55 m, perpendicular height 54.79 m ⇒ angle with ground $\theta$ satisfies $\cos\theta=\tfrac{54.79}{55}\Rightarrow\theta\approx4.1^\circ$ (tower leans $~4^\circ$ ).
• Garden swing: pole 2 m, chain 1.8 m, maximum swing angle $50^\circ$ .
– Closest safe distance $d=1.8\sin50^\circ\approx1.38\text{ m}$ from wall.
– Without safety stop, seat would hit wall at height $h=2-1.8\cos50^\circ\approx0.84\text{ m}$ above ground.

Summary of Key Identities & Tips

• Primary ratios: $\sin,\cos,\tan$ — always attach the angle.
• Reciprocal identities: $\csc\theta=1/\sin\theta,\;\sec\theta=1/\cos\theta,\;\cot\theta=1/\tan\theta.$
• Pythagorean identities: $\sin^{2}\theta+\cos^{2}\theta=1$ , $1+\tan^{2}\theta=\sec^{2}\theta$ , $1+\cot^{2}\theta=\csc^{2}\theta.$
• CAST diagram determines sign of ratio.
• When stuck:
– Draw a diagram.
– Label sides relative to chosen angle.
– Choose appropriate ratio (SOH-CAH-TOA).
– Keep calculator in degrees.
• Special angles table should be memorised for quick non-calculator work.
• Always state answer units (metres, degrees) & rounding instruction (1 dec place or 2 digits).