Right triangles and trigonometry

What the exam asks

About 15% of scored Math questions come from Geometry and Trigonometry, and right triangles and trigonometry usually account for one or two of them. Typical tasks:

• Find a missing side with the Pythagorean theorem or a Pythagorean triple.
• Use the $30^\circ$-$60^\circ$-$90^\circ$ and $45^\circ$-$45^\circ$-$90^\circ$ side patterns (both are on the reference sheet).
• Compute or use the sine, cosine and tangent of an acute angle, often given as a ratio such as $\tan A = \frac{5}{12}$.
• Apply $\sin x^\circ = \cos(90 - x)^\circ$, typically as “$\sin(a^\circ) = \cos(b^\circ)$; find ”.
• Convert between degrees and radians, and read sine and cosine from the unit circle.

Similar triangles tie it all together: every right triangle with the same acute angle has the same trig ratios.

Core ideas

Pythagorean theorem and triples

For legs $a$ and $b$ and hypotenuse $c$: $a^2 + b^2 = c^2$. The hypotenuse is always the longest side, opposite the right angle.

Spotting a triple turns a square-root calculation into a two-second step.

Special right triangles

A square’s diagonal makes two $45^\circ$-$45^\circ$-$90^\circ$ triangles. An equilateral triangle’s altitude makes two $$-- triangles.

An isosceles right triangle with leg $s$ has perimeter $s\left(2 + \sqrt{2}\right)$. Given a perimeter such as $$, factor it as to read off .

SOH-CAH-TOA

For an acute angle $\theta$ in a right triangle:

“Opposite” and “adjacent” depend on which angle you stand at. A ratio such as $\sin A = \frac{7}{25}$ gives the triangle’s shape (sides $7$, $24$ and , all times some ); one actual length gives the .

The complementary-angle identity

In right triangle $ABC$ with the right angle at $C$, angles $A$ and $B$ are complementary, and the side opposite $A$ is adjacent to $B$. So:

• $\sin A = \cos B$ and $\cos A = \sin B$; in general, $\sin x^\circ = \cos(90 - x)^\circ$.

In radians the identity reads $\sin\theta = \cos\left(\frac{\pi}{2} - \theta\right)$.

Radians and the unit circle

$\pi$ radians equal $180^\circ$. Multiply by $\frac{\pi}{180}$ to go from degrees to radians, and by to go back.

On the unit circle, the point at angle $\theta$ (measured counterclockwise from the positive $x$-axis) is $(\cos\theta, \sin\theta)$. For a point $(x, y)$ at distance from the origin, and . Find the reference angle with a special triangle, then use the quadrant to fix the signs.

Worked examples

Exam technique

• Sketch and label. If there’s no figure, draw the right triangle and mark the right angle, the given angle, and which sides are opposite, adjacent and hypotenuse.
• Triples first, radicals second. Check for multiples of $3$-$4$-$5$, $5$-$12$-$13$, -- and -- before squaring anything.

Common mistakes

Quick recap

• $a^2 + b^2 = c^2$; memorize the common triples and their multiples.
• $45^\circ$--: , , . --: , , .