Calculate Tangent from Sine and Cosine

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Find tan(π/4) exactly, using sinθ=√2/2 and cosθ=√2/2. Candidate answers: (A) 1 (B) −1 (C) −√3 (D) √3.
• Independent check inputs: Find tan(π/6) exactly, using sinθ=1/2 and cosθ=√3/2. Candidate answers: (A) 1 (B) −1 (C) √3 (D) √3/3.

Learn the idea

Radians measure an angle by arc length relative to radius. On the unit circle, cosine and sine are horizontal and vertical coordinates, so quadrant signs and special triangles support exact values beyond acute angles. Tangent is sine divided by cosine wherever cosine is nonzero. Special reference angles supply exact sine and cosine ratios; division and quadrant signs determine the corresponding tangent.

Here is the method on the supplied teaching case. Tangent is sine divided by cosine. Substitute (√2/2)/(√2/2) and simplify to 1. Cosine is nonzero; the sign follows the quotient of the supplied signs.

A ratio with zero cosine is undefined, but these supplied angles avoid that case. Do not copy the sine sign alone; divide its sign by cosine’s sign to determine tangent’s sign.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Find tan(5π/3) exactly, using sinθ=−√3/2 and cosθ=1/2. Candidate answers: (A) 1 (B) −1 (C) −√3 (D) √3. Show the mathematical reason, not only the selected answer. Expected result: −√3. Tangent is sine divided by cosine. Substitute (−√3/2)/(1/2) and simplify to −√3. Cosine is nonzero; the sign follows the quotient of the supplied signs.

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