Trigonometry

Trigonometry connects angles, side ratios, circles, waves, vectors, and geometric measurement.

Trigonometric functions connect angle, unit-circle coordinates, and right-triangle ratios. A radius at angle theta meets the unit circle at a labelled cosine-sine coordinate. Its horizontal and vertical projections form a reference triangle that links circular and triangle definitions.

Trigonometry begins with triangle ratios and grows into a language for rotation, direction, and repeating change.

Right-triangle foundation

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

\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},

\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},

\tan\theta=\frac{\text{opposite}}{\text{adjacent}}.

The memory aid SOH–CAH–TOA helps select a ratio after the triangle is labelled relative to the chosen angle.

Solving right triangles

To find a side, write a ratio containing the known side and target, then solve the equation. To find an angle from two sides, use an inverse trig function.

Check calculator degree/radian mode, keep full precision until the end, and verify that the hypotenuse is longest.

Worked example

In a realistic survey, add eye height if the sight line begins above ground.

Unit-circle meaning

On the unit circle, the point at angle $\theta$ is

(\cos\theta,\sin\theta).

This defines sine and cosine for all real angles, explains quadrant signs, and shows why the functions repeat.

Common mistakes

Choosing a trig ratio before selecting the reference angle. Side names are relative.

Using inverse trig to find a side. Inverse functions usually recover angles.

Mixing degrees and radians. Match notation and calculator mode.

Applying right-triangle ratios directly to a non-right triangle. Use appropriate laws or an altitude.

Treating a model as exact outside its assumptions. Interpret context.

Quick self-check

Is the triangle right or non-right?
Which sides and angles are known and requested?
Does the chosen ratio/law contain that information?
Is calculator mode correct and precision delayed?
Do signs, quadrants, and units fit the geometry?
Is the final answer reasonable and within the model's domain?