Right-angled triangle trigonometry
Trigonometry is the study of the relations between the sides and angles of triangles. The word “trigonometry” is derived from the Greek words trigono, meaning “triangle”, and metro, meaning “measure”. Though the ancient Greeks, such as Hipparchus and Ptolemy, used trigonometry in their study of astronomy between roughly 150 B.C. – A.D. 200, its history is much older. For example, the Egyptian scribe Ahmes recorded some rudi- mentary trigonometric calculations (concerning ratios of sides of pyramids) in the famous Rhind Papyrus sometime around 1650 B.C.
The first use of the idea of ‘sine’ in the way we use it today was in the work Aryabhatiyam by Aryabhatta, in A.D. 500. Aryabhatta used the word ardha-jya for the half-chord, which was shortened to jya or jiva in due course. When the Aryabhatiyam was translated into Arabic, the word jiva was retained as it is. The word jiva was translated into sinus, which means curve, when the Arabic version was translated into Latin. Soon the word sinus, also used as sine, became common in mathematical texts throughout Europe. Aryabhata An English Professor of astronomy Edmund Gunter (1581–1626), first used the abbreviated notation ‘sin’.
The origin of the terms ‘cosine’ and ‘tangent’ was much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhatta called it kotijya. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir Jonas Moore first used the abbreviated notation ‘cos’.
Trigonometry is distinguished from elementary geometry in part by its extensive use of certain functions of angles, known as the trigonometric functions.
(a) An angle is acute if it is between 0◦ and 90◦.
(b) An angle is a right angle if it equals 90◦.
(c) An angle is obtuse if it is between 90◦ and 180◦.
(d) An angle is a straight angle if it equals 180◦.
Relationships between angles
(a) Two acute angles are complementary if their sum equals 90◦. In other words, if 0◦ ≤ ∠A,∠B≤90◦ then ∠A and ∠B are complementary if ∠A+∠B=90◦.
(b) Two angles between 0◦ and 180◦ are supplementary if their sum equals 180◦. In other words, if 0◦ ≤∠A,∠B≤180◦ then ∠A and ∠B are supplementary if ∠A+∠B=180◦.
(c) Two angles between 0◦ and 360◦ are conjugate (or explementary) if their sum equals 360◦. In other words, if 0◦ ≤ ∠ A , ∠ B ≤ 360◦ then ∠ A and ∠ B are conjugate if ∠ A+∠ B = 360◦ .
In a right triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called its legs. For example, in this triangle the right angle is C, the hypotenuse is the line segment AB, which has length c, and BC and AC are the legs, with lengths a and b, respectively. The hypotenuse is always the longest side of a right triangle.
By knowing the lengths of two sides of a right triangle, the length of the third side can be determined by using the Pythagorean Theorem.
Trignometric functions of an acute angle
Consider a right triangle △ABC, with the right angle at C and with lengths a, b, and c, as in the figure on the right. For the acute angle A, call the leg BC its opposite side, and call the leg AC its adjacent side. The hypotenuse of the triangle is the side AB. The ratios of sides of a right triangle occur often enough in practical applications to warrant their own names, so we define the six trigonometric functions of A as follows:
Name of function Abbreviation = Definition
sine A Sin A = opposite side/hypotenuse = a/c
cosine A Cos A = adjacent side/hypotenuse = b/c
tangent A Tan A = opposite side/adjacent side = a/b
cosecant A Cosec A = hypotenuse/opposite side = c/a
secant A Sec A = hypotenuse/adjacent side = c/b
cotangent A Cot A = adjacent side/opposite side = b/a
We usually use the abbreviated names of the functions.
From the above table notice that Sin A and Cosec A, Cos A and Sec A and Tan A and Cot A are reciprocals.
Cosec A = 1/Sin A Sin A = 1/Cosec A Cos A = 1/Sec A
Sec A = 1/Cos A Tan A = 1/Cot A Cot A = 1/Tan A
Now with the same triangle we define the six trigonometric fuctions of B as follows:
Name of function Abbreviation = Definition
Sine B Sin B = opposite side/hypotenuse = b/c
Cosine B Cos B = adjacent side/hypotenuse = a/c
Tangent B Tan B = opposite side/adjacent side = b/a
Cosecant B Cosec B = hypotenuse/opposite side = c/b
Secant B Sec B = hypotenuse/adjacent side = c/a
Cotangent B Cot B = adjacent side/opposite side = a/c
We can noice the connections between sine and cosines, secants and cosecants and tangent and cotangent. This is called cofunction theorem.
Cofunction theorem : If A and B are complementary acute angles( A+B=90◦) of a right-angled triangle ABC, then the following relation hold:
Sin A = Cos B Cos A = Sin B Sec A = Cosec B
Cosec A = Sec B Cot A = Tan B Tan A = Cot B
We say that pair of functions (sin,cos), (sec,cosec) and (tan,cot) are cofunctions.
Trigonometric functions of any angle
To define the trigonometric functions of any angle – including angles less than 0◦ or greater than 360◦ – we need a more general definition of an angle. We say that an angle is formed by rotating a ray OA about the endpoint O (called the vertex), so that the ray is in a new position, denoted by the ray OB. The ray OA is called the initial side of the angle, and OB is the terminal side of the angle.
We denote the angle formed by this rotation as ∠AOB, or simply ∠O, or even just O. If the rotation is counter-clockwise then we say that the angle is positive, and the angle is negative if the rotation is clockwise. One full counter-clockwise rotation of OA back onto itself (called a revolution), so that the terminal side coincides with the initial side, is an angle of 360◦; in the clockwise direction this would be −360 . Not rotating O A constitutes an angle of 0 .
More than one full rotation creates an angle greater than 360◦. For example, notice that 30◦ and 390◦ have the same terminal side since 30 + 360 = 390.
Trigonometric function of any sngle can also be defined on cartesian coordinates. The xy-coordinate plane consists of points denoted by pairs (x,y) of real numbers. The first number, x, is the point’s x coordinate, and the second number, y, is its y coordinate. The x and y coordinates are measured by their positions along the x-axis and y-axis, respectively, which determine the point’s position in the plane. This divides the x y-coordinate plane into four quadrants (denoted by QI, QII, QIII, QIV), based on the signs of x and y.
These are the signs of trigonometric functions in different coordinates. Well if anyone find it difficult to memorize all these signs then you can memorize a formula which is “all school to college” this means all trigonometric functions are positive in QI then sin is positive in QII and if sign is positive its reciprocal cosec will also be positive then in QIII t for tan is positive and its reciprocal cot is also positive and finally in QIV cos is positive and its reciprocal sec is positive as well. This is ALL School To College.
Now the following table summarizes the values of trigonometric functions between 0◦ and 360◦
Here we have the values of sine cosine and tangent from 0◦ and 360◦. And we can figure out the values of cosecant secant and cotangent by using cofunction theorem and reciprocal formula. Since 360◦ represents one full revolution, the trigonometric function values repeat every 360◦. For example, sin 360◦ = sin 0◦, cos 390◦ = cos 30◦, tan 540◦ = tan 180◦, sin (−45◦) = sin 315◦, etc. In general, if two angles differ by an integer multiple of 360◦ then each trigono- metric function will have equal values at both angles. Angles such as these, which have the same initial and terminal sides, are called coterminal.
Rotations and Reflections of Angles
To rotate an angle means to rotate its terminal side around the origin when the angle is in standard position. For example, suppose we rotate an angle θ around the origin by 90◦ in the counterclockwise direction.
In figure an angle θ in QI which is rotated by 90◦, resulting in the angle θ+90◦ in QII. Notice that the complement of θ in the right triangle in QI is the same as the supplement of the angle θ + 90◦ in QII, since the sum of θ, its complement, and 90◦ equals 180◦. This forces the other angle of the right triangle in QII to be θ. Thus, the right triangle in QI is similar to the right triangle in QII, since the triangles have the same angles. The rotation of θ by 90◦ does not change the length r of its terminal side, so the hypotenuses of the similar right triangles are equal, and hence by similarity the remaining corresponding sides are also equal. Using Figure to match up those corresponding sides shows that the point (−y,x) is on the terminal side of θ+90◦ when(x,y) is on the terminal side of θ. Hence, by definition,
sin(θ+90◦) = cosθ
cos(θ+90◦) = −sinθ
tan(θ+90◦) = −cotθ.
You can notice that that value of cos and tan are in negative because angle (θ+90◦) lies in QII and in QII only sin and cosec are positive and cos and all other trigonometric functions are negative that’s why the value of cos and tan are negative.
We now consider rotating an angle θ by 180◦. Notice from Figure a and b that the angles θ ± 180◦ have the same terminal side, and are in the quadrant opposite θ. Since (−x, − y) is on the terminal side of θ ± 180◦ when (x, y) is on the terminal side of θ, we get the following relations, which hold for all θ.
sin(θ±180◦) = −sin θ
cos(θ±180◦) = −cos θ
tan(θ±180◦) = tanθ
A reflection is simply the mirror image of an object. For example, in Figure the original object is in QI, its reflection around the y-axis is in QII, and its reflection around the x-axis is in QIV. Notice that if we first reflect the object in QI around the y-axis and then follow that with a reflection around the x-axis, we get an image in QIII. That image is the reflection around the origin of the original object, and it is equivalent to a rotation of 180◦ around the origin. Notice also that a reflection around the y-axis is equivalent to a reflection around the x-axis followed by a rotation of 180◦ around the origin.
Applying this to angles, we see that the reflection of an angle θ around the x-axis is the angle −θ, as in Figure. So we see that reflecting a point (x, y) around the x-axis just replaces y by −y. Hence:
sin(−θ) = −sinθ
cos (−θ) = cos θ
tan(−θ) = −tanθ
Notice that the cosine function does not change in formula because it depends on x, and not on y, for a point (x, y) on the terminal side of θ. In general, a function f (x) is an even function if f (−x) = f (x) for all x, and it is called an odd function if f (−x) = − f (x) for all x. Thus, the cosine function is even, while the sine and tangent functions are odd. Replacing θ by −θ in formulas above, then using formulas given, gives:
Note that formulas extend the Cofunction Theorem to all θ, not just acute angles. Similarly, formulas give:
sin (180◦ − θ) = sin θ
cos(180◦−θ) = −cosθ
tan(180◦−θ) = −tanθ
Notice that reflection around the y-axis is equivalent to reflection around the x-axis (θ → −θ) followed by a rotation of 180◦ (−θ → −θ + 180◦ = 180◦ − θ), as in Figure. It may seem that these geometrical operations and formulas are not necessary for evaluating the trigonometric functions since we could just use a calculator. However, there are two reasons for why they are useful. First, the formulas work for any angles, so they are often used to prove general formulas in mathematics and other fields. Second, they can help in determining which angles have a given trigonometric function value.