Have you ever observed a clock? When the minute hand moves from 12 to 6, it forms a 180∘ angle. Even in one complete rotation, the hand forms a 360∘ angle.
In mathematics, we need to understand trigonometric values for angles like these. Not just limited to acute angles in right triangles.
Notice that as the point moves around the circle, the x and y coordinates can be positive or negative. This is what causes the signs of trigonometric functions to change.
Sign Change Visualization
Observe how sin, cos, and tan values change as the angle passes through each quadrant.
Sin (0°) = 0.00Cos (0°) = 1.00Tan (0°) = 0.00
0.00 Radian
Signs in each quadrant:
Quadrant
Angle Range
x
y
sin
cos
tan
I
0∘<θ<90∘
To avoid confusion, we can remember this with "All Students Take Calculus". In quadrant I All are positive, in quadrant II only sin is positive, in quadrant III only tan is positive, in quadrant IV only cos is positive.
A reference angle is an acute angle (0∘ to 90∘) formed between the terminal side of an angle and the nearest x-axis. This concept allows us to use trigonometric values of acute angles that we've already memorized.
Understanding Reference Angle
Notice the acute angle formed with the x-axis as the angle changes.
Sin (135°) = 0.71Cos (135°) = -0.71Tan (135°) = -1.00
Determine the values of sin315∘, cos315∘, and tan315∘.
Calculate sin(−60∘)+cos210∘−tan(−135∘).
If sinθ=53 and θ is in quadrant II, determine cosθ and .
Simplify sin840∘⋅cos(−330∘).
A windmill rotates 1050∘ from its initial position. If the initial position of the blade is on the positive x-axis, determine the coordinates of the blade tip on the unit circle after this rotation.
