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Trigonometric Ratios and Their Tricks

Trigonometric ratios (like sine, cosine, and tangent) relate the angles of a right triangle to the ratios of its sides.

Quadrants and Signs of Trigonometric RatiosThe value (sign) of a trigonometric ratio depends on the angle’s quadrant:
Quadrant I (0° to 90°): All ratios are positive.
Quadrant II (90° to 180°): Sine is positive, the rest are negative.
Quadrant III (180° to 270°): Tangent is positive, the rest are negative.
Quadrant IV (270° to 360°): Cosine is positive, the rest are negative��.

Odd and Even Multiples:
Odd-Even Identities
Even functions satisfy [ f(-x) = f(x) ]:
Example, cosine and secant (cos(-x) = cos(x), sec(-x) = sec(x)).
Odd functions satisfy [ f(-x) = -f(x) ]:
Example, sine, tangent, cosecant, cotangent (sin(-x) = -sin(x); tan(-x) = -tan(x))���

Odd/Even Multiple Angle Concept For angles that are odd or even multiples of 90° ([ n \times 90° ]), trigonometric function values transform using reference angles and sign rules:

When the angle is an even multiple of 90° (like 0°, 180°, 360°), the trigonometric ratio keeps its original function (e.g., sin(180° + θ) = -sin θ).

When the angle is an odd multiple of 90° (like 90°, 270°), the function changes (e.g., sin(90° + θ) = cos θ, tan(90° + θ) = -cot θ), depending on the quadrant��.

Practical Example
For sin(270° + θ):
270° is an odd multiple of 90°, in the 4th quadrant, and for sine it becomes:
sin(270° + θ) = -cos θ.

These rules make it easier to quickly determine trigonometric values for any angle, using quadrant sign rules and the transformation of the trigonometric function based on the angle’s multiple of 90°��.

"Master Trigonometric Ratios Easily! 🚦
Learn quick tricks to remember sin, cos, tan values, understand quadrant signs, and master odd/even multiple rules for any angle. Make trigonometry simple and fun with these life-saving tips!"


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