Summary
Obtain sin(90° − x):
The vertical side (sine) of the big angle matches the horizontal side (cosine) of the small triangle.In the 1st Quadrant, sine is on the positive side of the axis, so the result must be positive.
Therefore: sin(90° − x) = cos(x).
Obtain cos(90° − x):
The horizontal side (cosine) of the big angle matches the vertical side (sine) of the small triangle.In the 1st Quadrant, cosine is on the positive side of the x-axis, so the result must be positive.Therefore: cos(90° − x) = sin(x).
Obtain tan(90° -+ x) and cot(90° −+ x):
Since we use the vertical 90° axis, Tangent swaps to Cotangent, and Cotangent swaps to Tangent.
In the 1st Quadrant, both sine and cosine are positive (+ / +), so the result must be positive.
Therefore, tan(90° − x) = cot(x) and cot(90° − x) = tan(x).
In the 2nd Quadrant, sine is positive but cosine is negative (+ / −), so tan and cos both must be negative.
Therefore, tan(90° + x) = cot(x) and cot(90° + x) = tan(x).
Obtain sin(90° −+ x):
The vertical side (sine) of the big angle matches the horizontal side (cosine) of the small triangle.In the 1st Quadrant, sine is on the positive side of the axis, so the result must be positive.
Therefore: sin(90° − x) = cos(x).
In the 2nd Quadrant, sine is on the positive side of the vertical axis, so the result must be positive.
Therefore: sin(90° + x) = cos(x).
Obtain cos(90° − x):
The horizontal side (cosine) of the big angle matches the vertical side (sine) of the small triangle.
In the 1st Quadrant, cosine is on the positive side of the x-axis, so the result must be positive.
Therefore: cos(90° − x) = sin(x).
In the 2nd Quadrant, cosine lies on the negative side of the x-axis, so the result must be negative.
Therefore: cos(90° + x) = −sin(x).
Obtain tan(90° -+ x) and cot(90° −+ x):
Since we use the vertical 90° axis, Tangent swaps to Cotangent, and Cotangent swaps to Tangent.
In the 1st Quadrant, both sine and cosine are positive (+ / +), so the result must be positive.
Therefore, tan(90° − x) = cot(x) and cot(90° − x) = tan(x).
In the 2nd Quadrant, sine is positive but cosine is negative (+ / −), so tan and cos both must be negative.
Therefore, tan(90° + x) = cot(x) and cot(90° + x) = tan(x).
Obtain sin(270 - x):
- The vertical side (sin) of the big angle matches the horizontal side (cosine) of the small 30° triangle.
- Since we are in the 3rd Quadrant, sine is in the negative side of the axis, then the result must be negative.
- Therefore: sin(240) = - cos(30°).
- The horizontal side (cosine) of the big angle matches the vertical side (sine) of the small 30° triangle.
- Since we are in the 3rd Quadrant, the cosine is on the negative side of the x-axis, then the result must be negative.
- Therefore: cos(240°) = - sin(30°).
- Since we use the vertical 270° axis, Tangent swaps to Cotangent, and Cotangent swaps to Tangent
- In the 3rd Quadrant, since both sine and cosine are negative (- / -), the result must be positive.
- Therefore, tan(240) = cot(30) and cot(240) = tan(30).