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Algebra · Graphs of functions

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Trigonometric graphs

The sine and cosine waves from 0° to 360° — where they start, peak and cross — why they never leave the band −1 to 1, and how wave symmetry finds the second solution of sin x = k.

Algebra · Graphs of functions

Trigonometric graphs

The sine and cosine waves from 0° to 360° — where they start, peak and cross — why they never leave the band −1 to 1, and how wave symmetry finds the second solution of sin x = k.

Why it works

Sine and cosine are WAVES: they repeat every 360°360° and stay forever in the band between −1-1 and 11.

y=sin⁡xy = \sin x, from 0°0° to 360°360°: starts at 00, climbs to its peak 11 at 90°90°, back to 00 at 180°180°, dips to −1-1 at 270°270°, returns to 00 at 360°360°. Crossings at 0°,180°,360°0°, 180°, 360°.

y=cos⁡xy = \cos x is the same wave slid left: it STARTS at its peak — cos⁡0°=1\cos 0° = 1 — falls to 00 at 90°90°, bottoms at −1-1 at 180°180°, and recovers to 11 at 360°360°. "Sine starts at 0, cosine starts at 1" settles which curve is which at a glance.

The band is a hard law. sin⁡x=1.2\sin x = 1.2 has NO solutions — the wave never leaves [−1,1][-1, 1], so a horizontal line at 1.21.2 misses it entirely. That also caps scaled waves: 3sin⁡x3\sin x lives in [−3,3][-3, 3], so its maximum is exactly 3.

One solution breeds another — by symmetry. The sine wave is symmetric about x=90°x = 90° on its first hump, so

sin⁡x=k  ⇒  xand180°−x\sin x = k \;\Rightarrow\; x \quad\text{and}\quad 180° - x

are both solutions: from sin⁡30°=0.5\sin 30° = 0.5, the second solution of sin⁡x=0.5\sin x = 0.5 in range is 180°−30°=150°180° - 30° = 150°. On the negative side the wave repeats the hump upside-down: sin⁡x=−0.5\sin x = -0.5 first holds at 180°+30°=210°180° + 30° = 210° (and again at 360°−30°=330°360° - 30° = 330°).

For cosine the mirror is the vertical line x=180°x = 180° — solutions pair as xx and 360°−x360° - x: from cos⁡60°=0.5\cos 60° = 0.5, the other solution of cos⁡x=0.5\cos x = 0.5 is 300°300°.

Reading the graph beats memorising rules. Sketch the wave, draw the horizontal line y=ky = k, and count/locate the crossings — the symmetry formulas are just what the picture says.