StopLearnStopLearnThe sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. (The word comes from the Latin sinus for gulf or bay, since, given a unit circle, it is the side of the triangle on which the angle opens.) In our case
sin A = opposite/hypotenuse = a/h
This ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar.
The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse: so called because it is the sine of the complementary or co-angle. In our case
cos A = adjacent/hypotenuse = b/h
The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side: so called because it can be represented as a line segment tangent to the circle, that is the line that touches the circle, from Latin linea tangens or touching line (cf. tangere, to touch). In our case
tan A = opposite/adjacent = a/b
The acronyms “SOHCAHTOA” (“Soak-a-toe”, “Sock-a-toa”, “So-kah-toa”) and “OHSAHCOAT” are commonly used mnemonics for these ratios.
Reciprocal functions
The remaining three functions are best defined using the above three functions.
The cosecant csc (A), or cosec (A), is the reciprocal of sin (A); i.e. the ratio of the length of the hypotenuse to the length of the opposite side:
csc A = 1/sin A = hypotenuse/opposite = h/a
The secant sec (A) is the reciprocal of cos (A); i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:
sec A = 1/cos A = hypotenuse/adjacent = h/b.
It is so called because it represents the line that cuts the circle (from Latin: secare, to cut).
The cotangent cot (A) is the reciprocal of tan (A); i.e. the ratio of the length of the adjacent side to the length of the opposite side:
cot A = 1/tan A = adjacent/opposite = b/a.
Apart from finding the trigonometric ratios of angles by construction, the tables of trigonometric ratios in the mathematical tables can be used. These tables consist of both the natural and logarithmic sine, cosine and tangent of angles between 00 and 900 at intervals of 61 or 0.10and having the difference column on the extreme right for intermediate values. Also the table of trigonometric ratio of angles measured in radians are given, for the moment we will consider the natural trigonometric ratios (sine, cosine and tangent) of angles measured in degrees. The tables are usually supplied to save time, since finding the trig ratios of angles or the angles whose trig ratios are given takes a long time.
Solution: from the sine table, to find sin 320, we look for 320 under 0’ which gives 0.5299. i.e. sin 320 = 0.5299.
Find sin 430 30’
Solution: for sin 430 30’, we look for sin 430 under sin 30’ and get 0.6884.
i.e. sin 430 30’ = 0.6884.
Introduction
We shall consider the graphs of the following functions: sin x and cos x. We usually put y = sin x, y = cos x, to be able to plot points and draw the graphs.
The graph graph of y = sin x for 00 ≤ x ≤ 3600
From the figure it is evident that the curve repeats itself every 360o or 2p. This fact is expressed by saying that the function has a period of 360o or 2p.
In symbols we write sin (x + n.360o) or sin (x + 2np), sin x = sin (x + n.360o) = sin (x + 2np), where n is any positive or negative integer. This infers that sin x varies and takes a complete ordered range of values once and that sin x is periodic has the period 2p. From the figure we observe that as x increases from 0o to 90o, sin x increases from 0 to 1 and as x increases from 90o to 180o, sin x decreases from 1 to 0.
[A function f(x) is periodic with period T if f (x+T) = f(x) for all values of x]
As x increases from 180o to 270o, sin x decreases from 0 to -1 and as x increases from 270o to 360o, sin x increases from -1 to 0. The maximum absolute value of sin x = 1.
The graph of y = cos x, 00 ≤ x ≤ 3600
In this page we are going to discuss about Graph of cosine function. The trigonometric function cosine is defined as the ratio of adjacent side to the hypotenuse. The value of cosine always lies between 1 and -1.Trignometric functions are cyclic functions. It repeats the shape again and again. Study of trigonometric functions are used in many other fields of physics, chemistry and biology like simple harmonic motion, electronics
From the graph it is clear that the curve repeats itself every 360o (2p rad). This fact is expressed by the statement that the function has a period of 360o (2p radians). In symbols we write cos (x + 360o . n) = cos (x + 2pn) = cos x where n is a positive or negative integer.
From the graph we also observe that cos x does not pass through the origin. The maximum and minimum values of cos x are +1 and -1 respectively. As x increases from 0o to 90o cos x decreases from 1 to 0, as x increases from 90o to 180o cos x decreases from 0 to -1, as x increases from 180o to 270o cos x increases from -1 to 0, as x increases from 270o to 360o cos x increases from 0 to 1. cos x is period and has a period 2p.
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