Use the Pythagorean identity to calculate , then use . If the quadrant is not given, alone generally produces two possible tangent values because it does not determine the sign of cosine.
Use the Pythagorean identity to calculate , then use . If the quadrant is not given, alone generally produces two possible tangent values because it does not determine the sign of cosine.
This method is IB Mathematics: Analysis and Approaches (AA) SL content from SL 3.6: Pythagorean identity and double angles. It is suitable for Paper 1 or Paper 2.
Start with
If , rearrange to obtain
Therefore,
The reason this works is that sine and cosine are the - and -coordinates of the same point on the unit circle. Squaring removes their signs, so the identity recovers only the magnitude of cosine. The location of the point, its quadrant, then restores the correct sign before tangent is formed.
Before calculating, identify the possible quadrant from the sign of sine and any interval given in the question. If sine is positive, lies in quadrant I or II; if sine is negative, it lies in quadrant III or IV.
| Quadrant | Signs of | Sign of |
|---|---|---|
| I | positive, positive | positive |
| II | positive, negative | negative |
| III | negative, negative | positive |
| IV | negative, positive | negative |
For example, suppose and lies in quadrant II. Then
Thus . Since cosine is negative in quadrant II, . Hence,
You can also recognize this as a scaled right triangle: opposite side , hypotenuse , and adjacent-side magnitude . This supports the algebra but does not replace the quadrant check, because triangle lengths are positive.
A common misconception is that taking the square root automatically gives positive cosine. In fact, , so the quadrant determines the sign. Without quadrant information, the example gives .
For the IB command term find, show the identity, rearrange for cosine, select the correct sign, and substitute into the tangent ratio. On Paper 1, retain exact fractions or surds rather than decimals.