Mathematics, five units

How do you solve a trigonometric equation over a full period?

Q: Why does my calculator only give me one solution?

Calculators evaluate inverse trigonometric functions as true mathematical functions, meaning they can only output one value for each input. For arcsin, output ranges from -90° to 90°; for arccos, from 0° to 180°. You must apply circle symmetry manually to find the remaining angles in the period.

Q: How do I know if I should use degrees or radians?

Check the domain given in the problem prompt. If the interval is stated as 0 <= x < 2pi, solve in radians and keep calculator modes set to radians. If the prompt writes 0° <= x < 360°, calculate in degrees.

Q: What should I do if an equation contains both sine and cosine?

Use standard trigonometric identities to rewrite the equation in terms of a single function. For example, replace sin^2(x) with 1 - cos^2(x) to form a quadratic equation in cosine, which you can then factor or solve using the quadratic formula.

Q: When do I need to discard a solution?

Discard solutions that fall outside the specified domain interval, or values that make any denominator zero in the original equation, such as values of x that make tan(x) or a rational fraction undefined.

Quick answer

To solve a trigonometric equation over a full period, find the first solution using the inverse trigonometric function on a calculator. Then find the second solution using symmetry: subtract from 180 degrees for sine, or subtract from 360 degrees for cosine. Tangent has only one solution per period. Finally, verify that all solutions lie within the required interval.

The topic explained

The logic behind solving across a period

A calculator inverse button returns only one principal value. When you solve a trigonometric equation over a full period, that single output is almost never the entire answer. Over a full turn of 360 degrees (or 2pi radians), sine and cosine functions hit every value in their range twice, while the tangent function repeats every 180 degrees (or pi radians).

Knowing how many solutions exist before you calculate provides an immediate sanity check. For an equation of the form sin(x) = c or cos(x) = c where -1 < c < 1, you must find two distinct solutions within each period.

Using symmetry to find the second angle

After obtaining the primary angle from the inverse function, use symmetry on the unit circle to calculate the second angle in the period. For sine, the matching y-value sits at the supplementary angle: 180° - x (or pi - x). For cosine, the matching x-value sits across the horizontal axis: 360° - x (or -x).

For example, solving sin(x) = 0.5 begins with arcsin(0.5) = 30°. Its supplement is 180° - 30° = 150°. Both 30° and 150° fall within the standard period [0°, 360°), providing the complete solution set.

Compound arguments and identities

When an equation features a compound argument, such as cos(2x) = 0.5, always solve for the entire inside expression first before dividing by the coefficient. Expanding the search range to two full periods (0° to 720°) captures all four solutions for 2x before you divide each by 2.

Never divide both sides of an equation by a trigonometric term like sin(x) or cos(x). If that function equals zero, dividing eliminates valid roots. Instead, move all terms to one side and factor out the common function.

Free printable worksheet

Solving Trigonometric Equations Across One Period

Solve each equation for x over the interval 0° <= x < 360°. Show the primary solution, the symmetry step, and the final list of angles.

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Solving Trigonometric Equations Across One PeriodName:

Solve: sin(x) = 0.5
Solve: cos(x) = 0
Solve: tan(x) = 1
Solve: 2cos(x) + sqrt(3) = 0
Solve: 2sin^2(x) - sin(x) = 0
Solve: cos(2x) = 0.5 for 0° <= x < 360°

Common questions

Why does my calculator only give me one solution?

Calculators evaluate inverse trigonometric functions as true mathematical functions, meaning they can only output one value for each input. For arcsin, output ranges from -90° to 90°; for arccos, from 0° to 180°. You must apply circle symmetry manually to find the remaining angles in the period.

How do I know if I should use degrees or radians?

Check the domain given in the problem prompt. If the interval is stated as 0 <= x < 2pi, solve in radians and keep calculator modes set to radians. If the prompt writes 0° <= x < 360°, calculate in degrees.

What should I do if an equation contains both sine and cosine?

Use standard trigonometric identities to rewrite the equation in terms of a single function. For example, replace sin^2(x) with 1 - cos^2(x) to form a quadratic equation in cosine, which you can then factor or solve using the quadratic formula.

When do I need to discard a solution?

Discard solutions that fall outside the specified domain interval, or values that make any denominator zero in the original equation, such as values of x that make tan(x) or a rational fraction undefined.

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