Sine, Cosine, and Tangent

Welcome, students! 🌟 In this lesson, you will explore three of the most important trigonometric functions: sine, cosine, and tangent. These functions help describe angles, triangles, circular motion, waves, and many patterns in the real world. By the end of this lesson, you should be able to explain what these functions mean, use them in calculations, and connect them to the larger study of trigonometric and polar functions.

Lesson objectives:

Explain the main ideas and terminology behind $\sin$, $\cos$, and $\tan$.
Apply AP Precalculus reasoning and procedures to solve problems.
Connect trigonometric functions to circular motion and polar ideas.
Summarize how sine, cosine, and tangent fit into the study of trigonometric and polar functions.
Use examples and evidence to support your understanding.

Think of a clock hand, a Ferris wheel, or a bouncing ball. All of these can be modeled with trigonometric functions because they repeat in regular patterns 🔄.

What Sine, Cosine, and Tangent Mean

In right triangle trigonometry, sine, cosine, and tangent connect an angle to ratios of side lengths. For an acute angle $\theta$ in a right triangle, the side opposite $\theta$ is the side across from the angle, the adjacent side is next to the angle, and the hypotenuse is the longest side.

The three main ratios are:

$$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$$

$$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$$

$$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$$

A common memory tool is SOH-CAH-TOA. This stands for Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, and Tangent = Opposite over Adjacent.

These ratios are not just formulas to memorize. They describe how an angle changes the shape of a triangle. If the angle gets larger, some ratios increase while others decrease. That relationship is a big reason trigonometry is useful in modeling real situations.

Example: Suppose a ramp makes a $30^\circ$ angle with the ground and has length $10$ feet. The height of the ramp is the side opposite the angle, so

$$\sin(30^\circ)=\frac{\text{height}}{10}$$

Since $\sin(30^\circ)=\frac{1}{2}$, the height is $5$ feet. This kind of reasoning is used in construction, engineering, and design 🏗️.

Unit Circle Connections

AP Precalculus goes beyond right triangles and uses the unit circle, which is a circle with radius $1$ centered at the origin. On the unit circle, each angle $\theta$ corresponds to a point with coordinates

$$\bigl(\cos(\theta),\sin(\theta)\bigr)$$

This is one of the most important ideas in trigonometry. It means cosine is the $x$-coordinate and sine is the $y$-coordinate of a point on the unit circle.

Because the radius is $1$, the distance formula simplifies the relationship between coordinates and trig values. On the unit circle, the hypotenuse is always $1$, so the ratios become direct coordinates.

Tangent can also be connected to the unit circle. When $\cos(\theta)\neq 0$,

$$\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$$

This formula is very useful because it shows tangent as a ratio of sine and cosine. It also explains why tangent is undefined when $\cos(\theta)=0$. For example, at $\theta=90^\circ$ and $\theta=270^\circ$, the cosine value is $0$, so tangent does not exist there.

Example: At $\theta=60^\circ$, the unit circle gives

$$\cos(60^\circ)=\frac{1}{2}$$

and

$$\sin(60^\circ)=\frac{\sqrt{3}}{2}$$

Then

$$\tan(60^\circ)=\frac{\sin(60^\circ)}{\cos(60^\circ)}=\sqrt{3}$$

This is a key exact value that appears often on exams.

Signs, Quadrants, and Reference Angles

The signs of sine, cosine, and tangent depend on the quadrant where the angle’s terminal side lies. This is important for determining exact values and checking whether answers make sense.

In Quadrant I, all three are positive.
In Quadrant II, $\sin(\theta)$ is positive, while $\cos(\theta)$ and $\tan(\theta)$ are negative.
In Quadrant III, $\tan(\theta)$ is positive, while $\sin(\theta)$ and $\cos(\theta)$ are negative.
In Quadrant IV, $\cos(\theta)$ is positive, while $\sin(\theta)$ and $\tan(\theta)$ are negative.

A reference angle is the acute angle formed between the terminal side of an angle and the $x$-axis. Reference angles help you find trig values for angles outside Quadrant I.

Example: Find $\sin(150^\circ)$. The reference angle is $30^\circ$. Since $150^\circ$ is in Quadrant II, sine is positive. Therefore,

$$\sin(150^\circ)=\sin(30^\circ)=\frac{1}{2}$$

Example: Find $\tan(210^\circ)$. The reference angle is $30^\circ$. Since $210^\circ$ is in Quadrant III, tangent is positive. So,

$$\tan(210^\circ)=\tan(30^\circ)=\frac{\sqrt{3}}{3}$$

These ideas help you move from memorizing a few values to understanding the whole system.

Using Sine, Cosine, and Tangent in Real Problems

Trigonometric ratios are often used to find missing side lengths or angles in right triangles. If you know one side and one angle, you can often solve for the rest.

Example: A tree casts a $12$-meter shadow, and the angle of elevation from the tip of the shadow to the top of the tree is $40^\circ$. Let $h$ be the tree’s height. Then

$$\tan(40^\circ)=\frac{h}{12}$$

Solving gives

$$h=12\tan(40^\circ)$$

Using a calculator,

$$h\approx 10.07$$

So the tree is about $10.1$ meters tall.

This type of problem shows why tangent is especially useful when the opposite side and adjacent side are known. If the hypotenuse is involved, sine or cosine may be better.

Example: A drone is flying $50$ feet above the ground. The line of sight from a person on the ground to the drone makes a $35^\circ$ angle with the ground. Let $d$ be the distance from the person to the drone. Then

$$\sin(35^\circ)=\frac{50}{d}$$

$$d=\frac{50}{\sin(35^\circ)}$$

This gives a distance of about $87.1$ feet.

In AP Precalculus, you should choose the trig function that matches the sides you know and the side you want to find. That choice is a big part of mathematical reasoning.

How These Functions Fit the Bigger Picture

Sine, cosine, and tangent are more than triangle tools. They are also functions that model periodic behavior, meaning they repeat over regular intervals. This is why they appear in sound waves, daylight patterns, ocean tides, and rotating objects 🌊.

A function like $y=\sin(x)$ repeats every $2\pi$ radians. Its graph is smooth and continuous, oscillating between $-1$ and $1$. The graph of $y=\cos(x)$ has the same range and period, but it starts at a different point. The graph of $y=\tan(x)$ behaves differently: it repeats every $\pi$ radians and has vertical asymptotes where it is undefined.

These graphs matter because AP Precalculus asks you to connect symbolic formulas, graphs, and real-world context. For example, if a Ferris wheel rises and falls in a repeating motion, sine or cosine can model the rider’s height over time.

Polar coordinates also connect to these functions. In polar form, a point is described using a distance from the origin and an angle. Since angles and circular motion are central to sine and cosine, these functions naturally support polar graphing and analysis.

For instance, a point with polar coordinates $(r,\theta)$ can be translated to rectangular coordinates using

$$x=r\cos(\theta)$$

and

$$y=r\sin(\theta)$$

That connection shows why sine and cosine are foundational in the broader topic of trigonometric and polar functions.

Common Mistakes to Avoid

One common mistake is mixing up opposite and adjacent. The labels depend on the chosen angle, not on the triangle as a whole. Another mistake is forgetting that tangent is undefined when the cosine is $0$.

A second mistake is confusing degrees and radians. In AP Precalculus, both are used, so always check the unit of the angle. For example, $\pi$ radians equals $180^\circ$.

A third mistake is ignoring the quadrant. If you know only a reference angle, you still need to determine the sign from the quadrant. That sign is part of the exact answer.

Careful use of notation matters too. Write $\sin(\theta)$, not $\sin\theta$ in a way that could be unclear. Good notation helps communicate your reasoning clearly.

Conclusion

Sine, cosine, and tangent are core ideas in AP Precalculus because they connect geometry, algebra, and real-world modeling. students, you should now understand that these functions begin with side ratios in right triangles, extend to the unit circle, and appear in periodic graphs and polar coordinates. They are powerful tools for finding missing measurements, analyzing motion, and interpreting patterns in the world around you. Mastering these functions gives you a strong foundation for the rest of trigonometric and polar functions. ✅

Study Notes

$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$
$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$
$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$
On the unit circle, a point has coordinates $\bigl(\cos(\theta),\sin(\theta)\bigr)$.
If $\cos(\theta)\neq 0$, then $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$.
Sine and cosine are periodic with period $2\pi$; tangent has period $\pi$.
Reference angles help find trig values outside Quadrant I.
Signs depend on the quadrant: all positive in Quadrant I, only sine in Quadrant II, only tangent in Quadrant III, only cosine in Quadrant IV.
Right-triangle trig is useful for finding missing sides and angles in real situations like ramps, shadows, and height measurements.
Polar coordinates use trig through $x=r\cos(\theta)$ and $y=r\sin(\theta)$.
Always check whether the angle is in degrees or radians before solving.