Sine and Cosine on the Same Axes

Plotted side-by-side, sine and cosine reveal a beautiful relationship: they have the exact same shape, but $\cos x$ is shifted to the left by $\pi/2$ (or equivalently, sine is shifted to the right by $\pi/2$). One is the other in disguise.

The Phase Relationship

These two formulas are equivalent:

$\cos x = \sin\!\left(x + \dfrac{\pi}{2}\right)$

$\sin x = \cos\!\left(x - \dfrac{\pi}{2}\right)$

Step-by-Step Solution

Given: $f(x) = \sin x$ and $g(x) = \cos x$ on $[0, 2\pi]$.

Find: Their values at the four axis-aligned angles, the relationship between them, and where they intersect.

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Step 1 — Tabulate at the cardinal angles.

• $x = 0$: $\sin 0 = 0$, $\cos 0 = 1$
• $x = \pi/2$ (90°): $\sin(\pi/2) = 1$, $\cos(\pi/2) = 0$
• $x = \pi$ (180°): $\sin\pi = 0$, $\cos\pi = -1$
• $x = 3\pi/2$ (270°): $\sin(3\pi/2) = -1$, $\cos(3\pi/2) = 0$
• $x = 2\pi$ (360°): $\sin 2\pi = 0$, $\cos 2\pi = 1$

Notice the pattern: when one is at a peak ($\pm 1$), the other is at a zero. They alternate.

Step 2 — Find where the two curves intersect.

We want $\sin x = \cos x$. Dividing both sides by $\cos x$ (assuming it's nonzero):

$\dfrac{\sin x}{\cos x} = 1 \;\Longrightarrow\; \tan x = 1$

Solving on $[0, 2\pi]$:

$x = \dfrac{\pi}{4} \;\;\text{or}\;\; x = \pi + \dfrac{\pi}{4} = \dfrac{5\pi}{4}$

Step 3 — Find the values at the intersection points.

At $x = \pi/4$: $\sin(\pi/4) = \cos(\pi/4) = \dfrac{\sqrt{2}}{2} \approx 0.707$. So they cross at $(\pi/4,\, \sqrt{2}/2)$.

At $x = 5\pi/4$: $\sin(5\pi/4) = \cos(5\pi/4) = -\dfrac{\sqrt{2}}{2} \approx -0.707$. They cross at $(5\pi/4,\, -\sqrt{2}/2)$.

Step 4 — Verify the phase shift formula at one point.

At $x = 0$: $\sin(0 + \pi/2) = \sin(\pi/2) = 1 = \cos 0$. ✓

At $x = \pi/2$: $\sin(\pi/2 + \pi/2) = \sin\pi = 0 = \cos(\pi/2)$. ✓

The identity $\cos x = \sin(x + \pi/2)$ holds at every $x$.

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Answer: Sine and cosine are identical curves shifted by $\pi/2$. On $[0, 2\pi]$, they intersect at:

• $\left(\dfrac{\pi}{4},\; \dfrac{\sqrt{2}}{2}\right) \approx (0.785,\; 0.707)$
• $\left(\dfrac{5\pi}{4},\; -\dfrac{\sqrt{2}}{2}\right) \approx (3.927,\; -0.707)$

Both have period $2\pi$, range $[-1, 1]$, and amplitude 1. Cosine "leads" sine by $\pi/2$ — wherever sine is, cosine was a quarter-period ago.

Try It

• Slide the point widget — watch both functions read out simultaneously.
• The two yellow dots mark intersection points $\sin x = \cos x$ at $x = \pi/4$ and $x = 5\pi/4$.
• Notice how one is at a peak whenever the other crosses zero.