Foundations of Trigonometry

Triangle Measurements and Side Ratios

The word trigonometry is built from Greek words for triangle and measure. Right triangles provide a natural starting point because their side ratios are determined by their angles. Trigonometry later extends these ideas to the unit circle, periodic functions, and many settings beyond triangles.

Recalling the Pythagorean Theorem

For a right triangle with legs $a$ and $b$ and hypotenuse $c$, the Pythagorean theorem states:

The hypotenuse is opposite the right angle. We will use this theorem to derive exact side ratios for special right triangles.

Ratios and Proportions

Trigonometric functions begin as ratios of side lengths. A ratio compares two quantities, while a proportion states that two ratios are equal.

Suppose a vertical ruler and a vertical tree stand on the same level ground and their shadows are measured at the same time. Parallel sunlight makes the two height-shadow triangles similar. If the ruler is $6 \text{ cm}$ tall, its shadow is $3 \text{ cm}$, and the tree's shadow is $150 \text{ cm}$, let the tree height be $h$:

The proportion works because the two triangles share the same two acute angles. Two shadows alone do not guarantee equal ratios.

Similar Triangles

Similar triangles have equal corresponding angles and proportional corresponding sides. Any one of these standard tests is sufficient to prove similarity:

Similar triangles have the same shape even when their sizes differ. Every corresponding length is multiplied by the same scale factor.

Applications of Similar Triangles

Use the common scale factor to find unknown lengths or preserve proportions:

1. Height measurement: Shadows or sight lines can form similar triangles for objects that are difficult to measure directly.

2. Scale drawings: A plan or model preserves proportions while every length is enlarged or reduced by one scale factor.

3. Geometry and construction: Indirect measurements can be checked through corresponding angles and proportional sides.

Trigonometric Ratios

Trigonometric ratios compare the sides of a right triangle relative to a chosen acute angle.

Imagine a right triangle with an angle $\theta$ (theta). We name its sides as:

• Hypotenuse: the side opposite to the right angle
• Opposite side: the side opposite to angle $\theta$.
• Adjacent side: the side adjacent to angle $\theta$ other than the hypotenuse.

From these three sides, we obtain six basic trigonometric ratios. For an acute angle in a nondegenerate right triangle, all three side lengths are positive, so every ratio below is defined:

1. Sine ($\sin$):

   $$\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}$$

2. Cosine ($\cos$):

   $$\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}$$

3. Tangent ($\tan$):

   $$\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}$$

4. Cosecant ($\csc$):

   $$\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite side}}$$

5. Secant ($\sec$):

   $$\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent side}}$$

6. Cotangent ($\cot$):

   $$\cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent side}}{\text{opposite side}}$$

Ratios Fixed by the Angle

Two right triangles with the same acute angle, for example $\theta=30^\circ$, also share the right angle. The AA test therefore proves that they are similar. Scaling every side by the same factor leaves each quotient of corresponding side lengths unchanged.

Simple example:

• Triangle $A$ has an opposite side of $3 \text{ cm}$ and a hypotenuse of $6 \text{ cm}$.
• Triangle $B$ has an opposite side of $5 \text{ cm}$ and a hypotenuse of $10 \text{ cm}$.

If we calculate $\sin \theta$ for both triangles:

• $\sin \theta$ for triangle $A$ is $\frac{3}{6} = 0.5$.
• $\sin \theta$ for triangle $B$ is $\frac{5}{10} = 0.5$.

Although the triangles have different sizes, $\sin \theta$ has the same value. The same reasoning applies to the other side ratios.

On the unit circle, the point corresponding to $\theta$ has coordinates $(\cos \theta,\sin \theta)$. This definition extends sine and cosine beyond the acute angles of a right triangle.

The right-triangle ratios and the unit-circle coordinates agree for acute angles, so the same values can be used in problems with triangles of different sizes.

Measuring Lengths and Modeling Periodic Change

Trigonometry connects angles with lengths and periodic change, so it appears in measurement, navigation, construction, and wave models. A shadow, a bearing, and a rotating wave all reduce to a ratio between two sides of a triangle.

Measuring Height Using Shadows

For nearby vertical objects on level ground, height divided by shadow length is the same when the shadows are measured at the same time. This follows from parallel sunlight and similar triangles.

More than $2200 \text{ years}$ ago, Eratosthenes estimated Earth's circumference by comparing the Sun's angle at two locations and combining the angular difference with the distance between those locations. Without the distance, the angular difference alone cannot determine the circumference.

Navigation

Ancient sailors used trigonometry for navigation by measuring the angle between the horizon and stars. Measuring that angle gave the ship's position at sea, where no landmark was available. The method works because a measured angle and a known distance determine a triangle completely.

Architecture and Construction

Architects use trigonometry to calculate roof slopes, tower heights, and many structural aspects of buildings. A roofer knows the horizontal run and the desired slope but not the length of the rafter, and a sine or tangent finds that missing length from the two known values.

Electronics and Waves

Sound waves, radio waves, and alternating electric currents can be modeled using sine and cosine functions. A wave repeats, so an angle that grows past a full turn describes the next cycle rather than a new direction. Sine and cosine carry that repetition for every periodic signal.

Measuring Height with Shadow Ratios

Choose nearby vertical objects on level ground and measure their shadows at nearly the same time. This keeps the solar angle effectively the same.

1. Measure a reference height $h_r$ and its horizontal shadow length $s_r$.

2. Calculate the common ratio $k=h_r/s_r$.

3. Measure another object's shadow length $s$ and estimate its height with $h=ks$.

4. If desired, calculate the Sun's elevation angle with $\theta=\arctan k$.

For example, if a $1.5 \text{ m}$ reference object casts a $2.0 \text{ m}$ shadow, then $k=0.75$. An object with an $8.0 \text{ m}$ shadow has the estimated height

If the ground slopes, an object is not vertical, or the measurement times differ substantially, this simple model no longer applies without adjustment.