4-5 Fundamental identities and finding the rest from one function

The three Pythagorean identities

• sin²θ + cos²θ = 1.
• 1 + tan²θ = sec²θ (divide the first by cos²θ).
• 1 + cot²θ = csc²θ (divide the first by sin²θ).

Method when one function is given

• Determine the quadrant from the sign of the given function and the stated condition, so the signs of the rest are fixed.
• Use sin² + cos² = 1 for the missing leg, then tan = sin/cos, and take reciprocals for csc, sec, cot.
• Geometric alternative: build a right triangle with the given ratio and place it in the correct quadrant to read the signs of x, y and r.
• Remember r > 0 always, so sin and cos carry the sign of y and x respectively.