Trigonometry examines the relationship between the sides of a triangle, more specifically, right triangles. A right triangle has a 90\u00b0 angle. The equations and ratios that describe the relationship between the sides of a triangle and its angles are trigonometric functions. In this particular article, we're going to explain one specific ratio: \"cos\" or cosine. But before we dive into cosine, let's take a look at the other ratios in trigonometry.<\/p>\n
When we define the trigonometric ratios, let us define a right-angled triangle with one of the angles named\u00a0x.\u00a0<\/em>This angle is 90\u00b0. You define the sides of a triangle as\u00a0a, b,\u00a0<\/em>and\u00a0c<\/em>\u00a0where\u00a0a<\/em>\u00a0is the side adjacent to\u00a0x<\/em>\u00a0and\u00a0b<\/em>\u00a0is the side opposite\u00a0x<\/em>.\u00a0c<\/em>\u00a0is the hypotenuse or the side opposite the right angle. There are six fundamental trigonometric functions.<\/p>\n Out of the six fundamental trigonometric functions, you will mostly be concerned with sin, cos, and tan.<\/p>\n You can define a cosine function using a right-angled triangle as defined above. However, you can use cosine in several other applications.<\/p>\n You can use the cosine using differential equations. The cos and sin are the two differentiable trig functions and they have a special relationship.<\/p>\n The above definitions are useful when solving differential equations. Both of the above expressions are solutions to the differential equation:<\/p>\n Trigonometric functions are also defined using power series. By applying the Taylor series to cosine, you can obtain another definition.<\/p>\n Euler had related the sine and cosine functions by the expression:<\/p>\n The\u00a0j\u00a0<\/em>in the above expressions refers to the imaginary unit, which is equivalent to the square root of (-1). Euler's expression or relationship is true for all complex values. This means that the formula is true for all real values of\u00a0x<\/em>.<\/p>\n If we add the above equations, we can find a concise expression for cos x in the complex domain as:<\/p>\n If the value of\u00a0x<\/em>\u00a0is real, you can write the expression as:<\/p>\n Since a full circle is 360\u00b0, you can express the cosine in different parts of a circle starting at 0\u00b0 up to 360\u00b0. In the first quadrant of a circle, angles from 0\u00b0 to 90\u00b0, the value of cos is positive. In the second quadrant with a range of angles from 90\u00b0 to 180\u00b0, the value of cos is negative. In the third quadrant with a range of angles from 180\u00b0 to 270\u00b0, the value of cos is still negative. In the fourth quadrant, with the range of angles from 270\u00b0 to 360\u00b0, the value of cos is positive.<\/p>\n Before I proceed, let me introduce a trigonometric identity. Trigonometric identities are relationships between the trigonometric functions which are true at all conditions. One of them is A right triangle has a sin of 0.866. Find the cosine of the angle.<\/p>\n Taking our trigonometric identity, we can rearrange the expression.<\/p>\n Since we know the value of\u00a0sin x<\/em>, let us substitute it for\u00a0sin2 x<\/em>\u00a0in the expression.<\/p>\n A right triangle (ABC) has a right angle at B. The length of the hypotenuse, AC, is 5cm and the side BC is 3 cm. Find the angle at C.<\/p>\n To refresh your memory, the cosine of an angle is\u00a0adjacent\/hypotenuse<\/em>. Let the angle at C be\u00a0x<\/em>.<\/p>\n x = 53<\/em>\u00b0<\/p>\n The angle at C is 53\u00b0.<\/p>\n The expression\u00a0cos-1<\/em>\u00a0means the cos inverse. It is the inverse of the cos function. If the cosine of an angle is\u00a0x<\/em>, then cos-1\u00a0x<\/em>\u00a0is the original angle.<\/p>\n Find the cosine of the following angles using our circle quadrants.<\/p>\n 660\u00b0 is bigger than a circle, which is 360\u00b0. But since an angle is a degree of turning, it means that the point has moved a full circle and then some. The full circle will not count since the angle of interest is the amount it turned from the starting point to the final point.<\/p>\n So Since 300\u00b0 falls within the fourth quadrant, it means that the value of cos is positive.<\/p>\n For the second question, 234\u00b0 is less than 360\u00b0 so the moving point has not moved a full circle. Also, 234\u00b0 falls within the third quadrant. Therefore, the value of cos is negative.<\/p>\n The third problem has a negative angle of -60\u00b0. Negative angles mean that the direction of movement is clockwise instead of the normal anticlockwise. So if you move clockwise 60\u00b0 you will end up in the fourth quadrant.<\/p>\n You can easily find the cos of an angle by looking it up in a cos table or by pressing\u00a0cos\u00a0<\/em>and the angle on a scientific calculator. On most scientific calculators, the cos-1 inverse function is a second function of the cos, usually on the same key. For such calculators, to use the cos inverse function, press SHIFT and COS on the calculator.<\/p>\n\n
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sin x = (opposite) \/ (hypotenuse) = b \/ c<\/code><\/em><\/li>\n<\/ul>\n<\/li>\n\n
cos x = (adjacent) \/ (hypotenuse) = a \/ c<\/code><\/li>\n<\/ul>\n<\/li>\n\n
tan x = (opposite) \/ (adjacent) = b \/ a<\/code><\/li>\n(b \/ c) \/ (a \/ c)<\/code>,<\/em> you will get b\/a<\/code> which is tan x.<\/em>\u00a0So\u00a0tan x<\/em> can be expressed as the ratio of sin to cos. tan x = sin x \/ cos x<\/code>.<\/li>\n<\/ul>\n<\/li>\n\n
csc x = 1 \/ sin x<\/code><\/li>\n<\/ul>\n<\/li>\n\n
sec x = 1 \/ cos x<\/code><\/li>\n<\/ul>\n<\/li>\n\n
cot x = 1 \/ tan x<\/code><\/li>\n<\/ul>\n<\/li>\n<\/ul>\nCosine Function<\/h2>\n
Defining Cosine using Differential Equations<\/h3>\n
cos x = ( d \/ dx ) sin x<\/code> <\/em>\u00a0\u00a0and<\/p>\n-sin x = ( d \/ dx ) cos x<\/code><\/p>\ny\u201d + y = 0<\/code><\/p>\nThe Power Series Expansion<\/h3>\n
cos x = 1 \u2013 ( x2 \/ 2! ) + ( x4 \/ 4! ) \u2013 ( x6 \/ 6! )<\/code>\u2026..<\/em><\/p>\nExponential Expression using Euler's Formula<\/h3>\n
ejx = cos x + j sin x<\/code><\/p>\ne-jx = cos x \u2013 j sinx<\/code><\/p>\ncos x = ( ejx + e-jx ) \/ 2<\/code><\/p>\ncos x = Re( ejx )<\/code><\/p>\nValues of Cosine in the Four Quadrants of a Circle<\/h3>\n
Examples of Using Cosines<\/h2>\n
cos2 x + sin2 x = 1<\/code>.\u00a0<\/em>Let's look at a few examples and apply this trigonometric identity.<\/p>\nExample 1<\/h3>\n
cos2 x = 1 \u2013 sin2 x<\/code><\/p>\ncos x = ( 1 \u2013 sin2 x )1\/2<\/code><\/p>\ncos x = ( 1 \u2013 sin2 x )1\/2<\/code><\/p>\ncos x = (1 \u2013 0.8662 )1\/2<\/code><\/p>\ncos x = 0.5<\/code><\/p>\nExample 2<\/h3>\n
cos x = 3 \/ 5<\/code><\/p>\nx = cos-1 ( 3 \/ 5 )<\/code><\/p>\ncos 60\u00b0 = 0.5<\/code><\/p>\ncos-1 0.5 = 60\u00b0<\/code><\/p>\nExample 3<\/h3>\n
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cos 660\u00b0 = cos ( 660 \u2013 360 )\u00b0 = cos 300\u00b0<\/code><\/p>\ncos 300\u00b0 = 0.5<\/code><\/p>\ncos 234\u00b0 = -0.588<\/code><\/em><\/p>\n-60\u00b0 = ( 360 \u2013 60 )\u00b0 = 300\u00b0<\/code><\/p>\ncos 300\u00b0 = 0.5<\/code><\/p>\nFinal Thoughts<\/h2>\n