Precalculus · Unit 3: Trigonometric and Polar Functions · 14 min read · Updated 2026-05-11
Sine, cosine, and tangent (right triangle) — AP Precalculus
AP Precalculus · Unit 3: Trigonometric and Polar Functions · 14 min read
1. Foundations of Right Triangle Trigonometry★★☆☆☆⏱ 3 min
Right triangle trigonometry provides the foundational definition of the three core trigonometric functions, based on constant side ratios of right-angled triangles. All right triangles with a congruent acute angle are similar, so the ratio of any two sides is constant for a given angle, regardless of triangle size. This definition applies to angles between $0^\circ$ and $90^\circ$ ($0$ and $\pi/2$ radians), before extending to all real numbers via the unit circle. Mastery here is required for all subsequent trigonometric work in AP Precalculus, making up 7-10% of total exam weight, appearing in both MCQ and FRQ sections.
2. Core Ratios and the SOHCAHTOA Definition★★☆☆☆⏱ 4 min
In any right triangle, sides are labeled *relative to the acute angle $\theta$ you are analyzing*, not the right angle. The hypotenuse is always the side opposite the right angle (and the longest side). The opposite side is the leg that does not touch $\theta$, while the adjacent side is the leg that touches $\theta$.
Exam tip: Always label your sides relative to the target angle, not the other acute angle. Mixing up opposite and adjacent is the most common error on basic ratio problems, and it is a common MCQ distractor.
3. Solving for Unknown Sides and Angles★★★☆☆⏱ 4 min
Once you know at least one side and one acute angle of a right triangle, you can find all unknown sides and angles using SOHCAHTOA, the Pythagorean theorem, and the fact that the two acute angles sum to $90^\circ$.
To solve for an unknown side: (1) Identify known/needed sides labeled relative to the known angle, (2) Select the trig ratio connecting known and unknown side, (3) Set up and rearrange the equation to solve.
To solve for an unknown acute angle: (1) Identify the two known sides labeled relative to the unknown angle, (2) Set up the matching trig ratio, then use the corresponding inverse trigonometric function ($\arcsin$, $\arccos$, $\arctan$) to solve for the angle.
Exam tip: Always check your calculator's angle mode before calculating trig or inverse trig values; mixing degrees and radians will always give an incorrect answer that matches one of the MCQ distractors.
4. Cofunction Identities for Right Triangles★★★☆☆⏱ 3 min
The two acute angles in a right triangle are complementary (they sum to $90^\circ$ or $\pi/2$ radians). For an acute angle $\theta$, the other acute angle is $90^\circ - \theta$, and the opposite and adjacent sides swap places relative to this new angle. This relationship gives us the cofunction identities.
\sin(\theta) = \cos(90^\circ - \theta)
\cos(\theta) = \sin(90^\circ - \theta)
\tan(\theta) = \cot(90^\circ - \theta)
These identities are useful for simplifying trigonometric expressions, connecting complementary angles, and saving time on the exam. The name "cosine" is short for "complementary sine", referencing this relationship.
Exam tip: If you see two angles that add up to $90^\circ$ (or $\pi/2$ radians) in an expression, always check for a cofunction identity before reaching for your calculator to save time.
5. AP-Style Worked Examples★★★☆☆⏱ 5 min
Common Pitfalls
Why: Students rush to label the "bottom side" as adjacent without confirming which angle they are solving for.
Why: Students forget to rearrange the equation correctly after setting up the ratio.
Why: Students keep the mode from a previous problem and forget to switch.
Why: Students memorize SOHCAHTOA first and forget its domain restriction.
Why: Students expect the hypotenuse to always be the horizontal or bottom side, not the side opposite the right angle.