Three Basic Trigonometric Ratios of Angles of any Magnitude and their Graphs

Three Basic Trigonometric Ratios of Angles

Sine (sin): In any right-angle triangle, the sine of an angle θ is the ratio of the length of the side opposite θ to the length of the hypotenuse.
Cosine (cos): In any right-angle triangle, the cosine of an angle θ is the ratio of the length of the adjacent side to θ to the length of the hypotenuse.
Tangent (tan): In any right-angle triangle, the tangent of an angle θ is the ratio of the sine of θ to the cosine of θ, which is equivalent to the ratio of the length of the side opposite θ to the length of the adjacent side.

Graphs and Properties of Trigonometric Ratios

Periodic Nature: All three trigonometric functions are periodic, which means they repeat their values in regular intervals. Sine and cosine functions are periodic with period 2π or 360°, whereas the tangent function is periodic with period π or 180°.
Sine Graph: The sine function produces a waveform that starts from 0, increases to 1 at π/2 radians (90°), decreases to 0 at π radians (180°), decreases further to -1 at 3π/2 radians (270°), and finally returns to 0 at 2π radians (360°).
Cosine Graph: The cosine function waveform starts from 1, decreases to 0 at π/2 radians (90°), decreases further to -1 at π radians (180°), increases back to 0 at 3π/2 radians (270°), and finally returns to 1 at 2π radians (360°).
Tangent Graph: The tangent graph starts from 0, and increases as it approaches π/2 radians (90°), at which point it becomes undefined (as the cosine in the denominator becomes 0). The function decreases as it moves past π/2 radians and approaches π radians (180°), where it returns to 0.

Applications of Trigonometric Ratios

Trigonometric ratios have widespread applications, including solving problems in fields like physics, engineering, computer science, etc.
They are particularly useful in modelling periodic phenomena, such as sound and light waves, or motion in cyclical paths.
Trigonometric ratios are also used in geometry, to solve problems involving triangles, circles, and other complex shapes. For example, they can help determine unknown side lengths or angles in right-angle triangles.

Inverse Trigonometric Functions

The inverse trigonometric functions, denoted as sin⁻¹, cos⁻¹ and tan⁻¹, are used to find the angle that corresponds to a given trigonometric ratio.
The range of sin⁻¹ and cos⁻¹ is between -90° to 90°, while the range of tan⁻¹ is between -∞ to ∞.