How do the y-intercepts of the parent sine and cosine graphs differ?

The sine graph $y = \sin(x)$ passes through the origin at $(0,0)$, while the cosine graph $y = \cos(x)$ starts at its maximum value $(0,1)$. This reflects the unit circle values where $\sin(0) = 0$ and $\cos(0) = 1$.

What is the difference between the period of $y = \tan(x)$ and $y = \sin(x)$?

The period of $y = \tan(x)$ is $180^{\circ}$ ($\pi$), whereas the period of $y = \sin(x)$ is $360^{\circ}$ ($2\pi$). This means the tangent function repeats its cycle twice as frequently as the sine function.

How do the ranges of $y = \cos(x)$ and $y = \tan(x)$ compare?

The range of $y = \cos(x)$ is restricted to $[-1, 1]$, meaning it oscillates between a fixed maximum and minimum. In contrast, $y = \tan(x)$ has a range of $(-\infty, \infty)$, meaning its values extend infinitely in both vertical directions.

What happens if a student solves $\sin(x) = k$ but only provides the value from the calculator?

The student will likely lose marks for missing the second solution within the required interval. Because trigonometric graphs are periodic and symmetrical, most horizontal lines intersect the curve twice per period.

What is the consequence of having a calculator in Degree mode when the question uses Radians?

The calculated values will be mathematically incorrect for the given context, leading to wrong intercepts and slopes on a sketch. Radians and degrees represent different scales for the x-axis, and results are not interchangeable without proper conversion.

Where do students often incorrectly draw the asymptotes for the tangent function?

Students often forget that asymptotes for $y = \tan(x)$ occur every $180^{\circ}$, starting at $90^{\circ}$ and $270^{\circ}$. A common error is placing them at the intercepts (like $0^{\circ}$ or $180^{\circ}$) instead of the undefined points.

Define the 'Amplitude' of a trigonometric function.

Amplitude is the distance from the equilibrium (center line) to the peak or trough of the wave. For the basic $y = \sin(x)$ and $y = \cos(x)$ graphs, the amplitude is exactly $1$.

What is the mathematical definition of a 'Periodic Function'?

A function $f$ is periodic if there exists a constant $P > 0$ such that $f(x + P) = f(x)$ for all values of $x$. The smallest such $P$ is called the period of the function.

What is a 'Vertical Asymptote' in the context of the tangent graph?

It is a vertical line $x = c$ where the function values grow without bound (approach $\infty$ or $-\infty$) as $x$ approaches $c$. For tangent, these occur at angles where the cosine is zero.

Why is the cosine graph symmetrical about the y-axis?

Because cosine is an even function, which means $\cos(-x) = \cos(x)$. In a unit circle, an angle of $x$ and $-x$ share the same x-coordinate, resulting in mirrored values on either side of the y-axis.

Library Podcasts

Courses

Referral & Rewards

Revision Notes

International A-Level

Graphs of Trigonometric Functions

Summary

Trigonometric graphs are periodic representations of the sine, cosine, and tangent functions on a coordinate plane. These graphs oscillate between specific values or approach asymptotes, providing a visual framework for solving equations and understanding cyclical behavior in mathematics and physics.

1. Definition & Core Concepts

A standard sine wave graph showing one full period from 0 to 360 degrees with peaks at 1 and troughs at -1.

2. Underlying Principles

Unit Circle Origin: Trigonometric graphs translate the circular motion of a point on a unit circle into a linear horizontal scale. The sine graph tracks the y-coordinate (vertical position), while the cosine graph tracks the x-coordinate (horizontal position) as the angle increases.
Even and Odd Symmetry: The cosine function is even, meaning it is reflected perfectly across the y-axis, such that $\cos(-x) = \cos(x)$ . The sine and tangent functions are odd, possessing rotational symmetry about the origin, where $\sin(-x) = -\sin(x)$ .
Continuity vs. Discontinuity: Sine and cosine are continuous functions defined for all real numbers, creating a smooth wave. The tangent function is discontinuous at every point where $\cos(x) = 0$ , resulting in separate branches separated by vertical asymptotes.

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

7. Connections & Extensions