Geometry, Trigonometry, and Complex Numbers
Master diagrams and identities under time pressure. Read units, angles, and constraints first; then compute with precision.
Quick Reference (Minimal Theory)
Use radius, not diameter.
\(\theta\) in radians.
Right triangles only.
For non-right triangles.
Check angle location.
Center \((h,k)\), radius \(r\).
Conjugate: \(\overline{z}=a-bi\)
For powers and roots of complex numbers.
Worked Examples (10) — Visual & Algebraic
Example 1 Medium
Find the area of a circle with diameter 10 cm.
Show solution
Radius \(r=5\). \(A=\pi r^2=25\pi\,\text{cm}^2\).
\(25\pi\,\text{cm}^2\)
Example 2 Ivy
Triangle with sides 7, 8, 9 — find its area.
Show solution
Semiperimeter \(s=12\). \(A=\sqrt{s(s-7)(s-8)(s-9)}=\sqrt{720}=12\sqrt5\).
\(12\sqrt5\)
Example 3 Ivy
Equation of circle centered at \((3,-2)\) with radius 5.
Show solution
\((x-3)^2+(y+2)^2=25\).
\((x-3)^2+(y+2)^2=25\)
Example 4 MIT
Find \(\sin\theta\) and \(\cos\theta\) for a right triangle with sides 5, 12, 13.
Show solution
\(\sin\theta=5/13\), \(\cos\theta=12/13\).
\(\sin\theta=5/13\), \(\cos\theta=12/13\)
Example 5 Ivy
Find angle \(A\) in triangle with sides 4, 5, 6 using Law of Cosines.
Show solution
\(\cos A=(b^2+c^2-a^2)/(2bc)=(25+36-16)/40=45/40=0.875\Rightarrow A\approx29^\circ\).
\(A\approx29^\circ\)
Example 6 Medium
Compute \((2+3i)(1-4i)\).
Show solution
\(=2-8i+3i-12i^2=14-5i\).
\(14-5i\)
Example 7 Ivy
Find modulus and argument of \(z=1-i\).
Show solution
\(|z|=\sqrt2\), \(\arg z=-45^\circ\).
\(|z|=\sqrt2\), \(\arg z=-45^\circ\)
Example 8 Ivy
Find equation of a line tangent to \(x^2+y^2=25\) at \((3,4)\).
Show solution
\(xx_1+yy_1=r^2\Rightarrow 3x+4y=25\).
\(3x+4y=25\)
Example 9 MIT
Prove \(\sin^2\theta+\cos^2\theta=1\) using a right triangle.
Show solution
\(\sin=op/hyp\), \(\cos=adj/hyp\) ⇒ sum \(=\tfrac{op^2+adj^2}{hyp^2}=1\).
Identity proven.
Example 10 Ivy
Find all solutions to \(\cos 2x=\tfrac12\) for \(0\le x<360^\circ\).
Show solution
\(2x=60^\circ,300^\circ\Rightarrow x=30^\circ,150^\circ\).
\(x=30^\circ,150^\circ\)
Practice Questions (20) — With Solutions
- Medium Find area of triangle with base 10 and height 6.
Solution
\(A=\tfrac12 bh=30\).
30
- Ivy Equation of circle with center \((-2,5)\) and radius 3.
Solution
\((x+2)^2+(y-5)^2=9\).
\((x+2)^2+(y-5)^2=9\)
- MIT Tangent to circle \(x^2+y^2=16\) at \((4,0)\).
Solution
\(xx_1+yy_1=r^2\Rightarrow 4x=16\Rightarrow x=4\).
\(x=4\)
- Medium Find slope of line perpendicular to \(2x-3y=6\).
Solution
Original slope \(=2/3\). Perpendicular slope \(=-3/2\).
\(-3/2\)
- Ivy Area of sector radius 6, central angle \(120^\circ\).
Solution
\(A=\tfrac{120}{360}\pi 6^2=12\pi\).
\(12\pi\)
- Medium \(\sin 30^\circ\), \(\cos 60^\circ\), \(\tan 45^\circ\).
Solution
\(\tfrac12,\tfrac12,1\).
\(1/2,\ 1/2,\ 1\)
- Ivy If \(\sin\theta=0.8\) and hypotenuse is 10, find the other sides.
Solution
Opposite = 8, adjacent = 6 (by \(a^2+b^2=10^2\)).
8 and 6
- MIT Solve \(2\sin x=1\) for \(0\le x<360^\circ\).
Solution
\(\sin x=0.5\Rightarrow x=30^\circ,150^\circ\).
\(30^\circ,150^\circ\)
- Ivy Prove \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\).
Solution
From SOH–CAH–TOA: \(\tan=op/adj=(op/hyp)/(adj/hyp)=\sin/\cos\).
Identity holds.
- Medium Evaluate \(\sin^2 45^\circ+\cos^2 45^\circ\).
Solution
Each is \(\tfrac{\sqrt2}{2}\). Square and add ⇒ 1.
1
- Medium Compute \(i^5\), \(i^6\), \(i^7\).
Solution
Cycle: \(i,i^2=-1,i^3=-i,i^4=1\). So \(i^5=i\), \(i^6=-1\), \(i^7=-i\).
\(i, -1, -i\)
- Ivy Simplify \(\dfrac{3+4i}{1-2i}\).
Solution
Multiply by conjugate: \(\frac{(3+4i)(1+2i)}{1+4}=\frac{-5+10i}{5}=-1+2i\).
\(-1+2i\)
- MIT If \(z=2(\cos30^\circ+i\sin30^\circ)\), compute \(z^3\).
Solution
De Moivre: magnitude \(8\), angle \(90^\circ\) ⇒ \(8i\).
\(8i\)
- Ivy Find conjugate of \(2-5i\).
Solution
\(2+5i\).
\(2+5i\)
- Medium Simplify \((1+i)^4\).
Solution
Expand or use polar; result \(-4\).
\(-4\)
Next Steps
Drill mixed sets combining geometry, trig, and complex numbers. Target: medium (<40s); advanced (<90s); accuracy ≥ 80%. Then continue to Full SAT Math Practice Tests.