4-6 Related angles: reduction to the reference angle

• The reference angle is the acute angle between the terminal side and the x-axis; the value of the function is that of the reference angle with the sign fixed by the quadrant.

The eight forms

• 180° − θ: sin(180−θ) = sinθ, cos(180−θ) = −cosθ, tan(180−θ) = −tanθ.
• 180° + θ: sin = −sinθ, cos = −cosθ, tan = tanθ.
• 360° − θ: sin = −sinθ, cos = cosθ, tan = −tanθ.
• −θ: sin = −sinθ, cos = cosθ, tan = −tanθ (cosine is even, sine and tangent odd).
• 90° − θ (co-functions): sin = cosθ, cos = sinθ, tan = cotθ.
• 90° + θ: sin = cosθ, cos = −sinθ, tan = −cotθ.
• 270° − θ: sin = −cosθ, cos = −sinθ, tan = cotθ.
• 270° + θ: sin = −cosθ, cos = sinθ, tan = −cotθ.
• Rule of thumb: for 90° and 270° the function changes to its co-function; for 0°, 180° and 360° it stays the same — the sign comes from the quadrant of the whole angle.