Search results
Results From The WOW.Com Content Network
The trigonometric functions cosine and sine of angle θ may be defined on the unit circle as follows: If (x, y) is a point on the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle θ from the positive x-axis, (where counterclockwise turning is positive), then = =.
The values of sine and cosine of 30 and 60 degrees are derived by analysis of the equilateral triangle. In an equilateral triangle, the 3 angles are equal and sum to 180°, therefore each corner angle is 60°. Bisecting one corner, the special right triangle with angles 30-60-90 is obtained.
To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed.
Fig. 1a – Sine and cosine of an angle θ defined using the unit circle Indication of the sign and amount of key angles according to rotation direction. Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane. [37]
The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series , or as the solutions of certain differential equations , allowing their extension to arbitrary positive and negative values and even to ...
Similar right triangles illustrating the tangent and secant trigonometric functions Trigonometric functions and their reciprocals on the unit circle. The Pythagorean theorem applied to the blue triangle shows the identity 1 + cot 2 θ = csc 2 θ, and applied to the red triangle shows that 1 + tan 2 θ = sec 2 θ.
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
The formulas for addition and subtraction involving a small angle may be used for interpolating between trigonometric table values: Example: sin(0.755) = (+) + () + () where the values for sin(0.75) and cos(0.75) are obtained from trigonometric table. The result is accurate to the four digits given.