Trigonometric functions Revision Notes

The sine function produces a smooth wave-like graph oscillating between -1 and 1. When plotting y = sin x for x from 0° to 360°, the graph starts at 0 (since sin 0° = 0), rises to a maximum of 1 at 90°, returns to 0 at 180°, falls to a minimum of -1 at 270°, and returns to 0 at 360°. This pattern forms one complete cycle of the sine wave.

Key points to plot are (0°, 0), (90°, 1), (180°, 0), (270°, -1), and (360°, 0). The graph is smooth and continuous with no breaks or asymptotes. The sine function’s range is limited to values between -1 and 1 inclusive, which is crucial when solving equations involving sine.

• Starts at 0 at 0°
• Maximum value 1 at 90°
• Returns to 0 at 180°
• Minimum value -1 at 270°
• Completes cycle at 360° with value 0

The cosine graph is similar in shape to the sine graph but shifted horizontally. For x from 0° to 360°, y = cos x starts at 1 (cos 0° = 1), decreases to 0 at 90°, reaches -1 at 180°, returns to 0 at 270°, and back to 1 at 360°. This also forms one complete wave cycle.

Plotting points such as (0°, 1), (90°, 0), (180°, -1), (270°, 0), and (360°, 1) helps sketch the curve. Like sine, cosine values range between -1 and 1, and the graph is continuous without breaks or asymptotes.

• Starts at 1 at 0°
• Decreases to 0 at 90°
• Minimum value -1 at 180°
• Returns to 0 at 270°
• Completes cycle at 360° with value 1

The tangent function behaves differently from sine and cosine. It has a period of 180°, so its pattern repeats twice between 0° and 360°. The graph passes through 0 at 0°, 180°, and 360°, but has vertical asymptotes where cosine is zero, specifically at 90° and 270°, because tan x = sin x / cos x and division by zero is undefined.

Between these asymptotes, the graph increases or decreases without bound, moving from negative infinity to positive infinity. This means the tangent function’s range is all real numbers, unlike sine and cosine.

When sketching, draw vertical dashed lines at 90° and 270° to indicate asymptotes, plot points such as (0°, 0), (45°, 1), (135°, -1), (225°, 1), and (315°, -1), and show the curve approaching the asymptotes.

• Period of 180°, repeats twice in 0° to 360°
• Vertical asymptotes at 90° and 270°
• Passes through 0 at 0°, 180°, 360°
• Range is all real numbers
• Graph increases/decreases without bound near asymptotes

When solving equations like sin x = k, cos x = k, or tan x = k for x between 0° and 360°, first consider the range of the function. For sine and cosine, k must be between -1 and 1 inclusive; otherwise, there is no solution.

For example, sin x = 2 has no solution because 2 is outside the sine function’s range.

For equations such as 2 cos x + 1 = 0, rearrange algebraically: 2 cos x = -1, so cos x = -1/2. Then find all angles between 0° and 360° where cosine equals -1/2. Using the cosine graph or unit circle, these are 120° and 240°. Always check the graph to confirm the number and approximate location of solutions within the domain.

• Check if the value is within the function’s range before solving
• Rearrange equations to isolate the trigonometric function
• Use the graph or unit circle to find all solutions in 0° to 360°
• Verify solutions by substitution or graph inspection

Always remember the domain restriction: only consider angles between 0° and 360° unless otherwise specified.

For sine and cosine, solutions often come in pairs within this domain due to the wave’s symmetry.

For tangent, be aware of the vertical asymptotes and the function’s periodicity of 180°, so solutions repeat every 180° within the domain. For example, if tan x = 1 at 45°, it also equals 1 at 225° (45° + 180°).

• Domain limited to 0° ≤ x ≤ 360°
• Sine and cosine have two solutions per cycle in this domain
• Tangent solutions repeat every 180°
• Check for extraneous solutions outside the function’s range