Introduction to Trigonometric
Trigonometry is the branch of mathematics that studies the relationships between angles and sides of triangles or that deal with angles, lengths, and heights of triangles and relations between different parts of circles and other geometrical figures. The Concepts of trigonometry are very useful in practical life as well as finds application in the field of engineering, astronomy, Physics, and architectural design.
These are in total 6 ratios that we study in trigonometry used to tell us about the triangle and Sine and Cos are two of them. We will discuss in detail about the Sin Cos ratio, formula and other concepts.
Trigonometric Ratios
In mathematics six Trigonometric Ratios for the right angle triangle are defined i.e Sine, Cosecant, Tangent, Cosecant, Secant respectively. These Trigonometric Ratios are real functions which relate an angle of a right-angled triangle to ratios of two of its side lengths. Sin and Cos are basic Trigonometric functions that tell about the shape of a right triangle.There are six Trigonometric Ratios, Sine, Cosine, Tangent, Cosecant, Secant and Cotangent and are abbreviated as Sin, Cos, Tan, Csc, Sec, Cot. These are referred to as ratios because they can be expressed in terms of the sides of a right-angled triangle for a specific angle \[\theta\].
Adjacent: It is the side adjacent to the angle being taken for consideration.
Opposite: It is the side opposite to angle being taken for consideration.
Hypotenuse: It is the side opposite to the right angle of the triangle or the largest side of the triangle.
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Sin θ = \[\frac{Perpendicular}{Hypotenuse}\] = \[\frac{Opposite}{Hypotenuse}\]
Cos θ = \[\frac{Base}{Hypotenuse}\] = \[\frac{Adjacent}{Hypotenuse}\]
Tan θ = \[\frac{Perpendicular}{Base}\] = \[\frac{Opposite}{Adjacent}\]
Cosec θ =\[\frac{Hypotenuse}{Perpendicular}\] = \[\frac{Hypotenuse}{Opposite}\]
Sec θ = \[\frac{Hypotenuse}{Base}\] = \[\frac{Hypotenuse}{Adjacent}\]
Cot θ = \[\frac{Base}{Perpendicular}\] = \[\frac{Adjacent}{Opposite}\]
Basic Identities of Sine and Cos
If \[A + B\] = \[90^{0}\],that is A and B are complementary to each other then:
\[Sin (A) = Cos (B)\]
\[Cos (A) = Sin (B)\]
If \[A + B\] = \[180^{0} \] then:
\[Sin (A) = Sin (B)\]
\[Cos (A) = - Cos (B)\]
\[ Cos^{2}(A) + Sin^{2}(A) = 1\]
Double and Triple Angles
The Double or Triple Angle ratios can be converted to Single angle ratio of Sin and Cos using the below mentioned formulae:
\[Sin2A = 2 SinA.CosA \]
\[Cos2A = Cos^{2}A – Sin^{2}A = 2Cos^{2}– 1 = 1 − 2Sin^{2}A \]
\[Sin3A = 3 SinA – 4 Sin^{3}A \]
\[Cos3A = 4 Cos^{3}A – 3 CosA \]
\[Sin^{3}A = 4 Cos^{3}A SinA – 4 CosA Sin^{3}A \]
\[Cos4 A = Cos^{4}A – 6 Cos^{2}A Sin^{2}A + Sin^{4}A \]
\[Sin^{2} A = \frac{(1 – Cos2A)}{2} \]
\[Cos^{2} A = \frac{(1 + Cos2A)}{2} \]
Sum and Difference of Angles
The Sin and Cos ratio of sum and difference of two angles can be converted two product using below mentioned identities:
Sin(A + B) = Sin(A) Cos(B) + Cos(A) Sin(B)
Sin(A − B) = Sin(A) Cos(B) − Cos(A) Sin(B)
Cos(A + B) = Cos(A) Cos(B) − Sin(A) Sin(B)
Cos(A − B) = Cos(A) Cos(B) + Sin(A) Sin(B)
Sin(A + B + C) = SinACosBCosC + CosASinBCosC + CosACosBSinC − SinASinBSinC
Cos(A + B + C) = CosACosBCosC − SinASinBCosC − SinACosBSinC − SinACosBSinC − CosASinBSinC
\[SinA + SinB = 2Sin \frac{(A + B)}{2}.Cos\frac{A - B}{2}\]
\[SinA – SinB = 2Sin\frac{A - B}{2}.Cos\frac{A + B}{2}\]
\[CosA + CosB = 2Cos \frac{(A + B)}{2}.Cos\frac{A − B}{2}\]
\[CosA + CosB = −2 Sin \frac{(A + B)}{2}.Sin\frac{A − B}{2}\]
In order to remember the trigonometric ratios values follow the given below steps:
We can the values for Sine ratios,i.e., 0, \[ \frac{1}{2}\], \[ \frac{1}{\sqrt{2}}\], \[ \frac{\sqrt{3}}{2}\], and 1 for angles \[0^{\circ} \],\[30^{\circ} \],\[45^{\circ} \], \[60^{\circ} \]and \[90^{\circ} \] and the Cos ratio will follow the exact opposite pattern i.e. 0, \[ \frac{1}{2}\], \[ \frac{1}{\sqrt{2}}\], \[ \frac{\sqrt{3}}{2}\], and 1 at\[90^{\circ} \],\[60^{\circ} \],\[45^{\circ} \],\[30^{\circ} \] and\[0^{\circ} \]. While the Tan will be the ratio of Sin and Cos ratio.
The value of Cosec, Sec and Cot is exactly the reciprocal of Sin, Cos and Tan.
Values of Trigonometric Ratios at Various Angles:
Trigonometry is a branch of mathematics and a sub-branch in algebra concerned with the measurement of specific functions of angles and their application to calculations. An example of trigonometry which is easy to understand is that of what architects use to calculate any particular distances.
Algebra and trigonometry are two major branches of mathematics. Algebra involves the study of math with specific formulas, rules, equations, and other variables. Trigonometry deals only with the triangles and their measurements.
The Six Main Functions of an Angle that are Commonly Used in Trigonometry are
sine (sin),
cosine (cos),
tangent (tan),
cotangent (cot),
secant (sec), and
cosecant (csc).
An easy and simple way to learn and understand Trigonometry is by studying all the basics of trigonometric angles and formulas by writing them down in a separate notebook which will be really useful to revise them before exams. Make sure you understand and study all the entire right-angle triangle concepts well so that you might compare any problems with a triangle before you try to solve them. The main thumb rule to score well in trigonometry is to learn your Pythagoras theorem with a whole heart. Keeping the Sine rule and Cosine rule at your fingertips will help you solve any type of problem in the examination. Finally, list down all the important identities and formulas of trigonometry in your mind and revision notes as well, and be thorough. Remember to learn how to use the trigonometry table in the necessary place.
FAQs on Sin Cos Tan Values
1. How can we Convert Cos to Sin?
Since the value of \[Cos(\theta) = Sin (90^{0} - (\theta)\]
Means that if \[\theta\] is equal to 35 degrees, then \[Cos\](\[45\]\[^{0}\]) = \[Sin (90^{0}\] – \[35^{0}\]) = \[Sin 55^{0}\].
Also \[Sin (\theta) = Cos(90^{0}) - (\theta)\]
2. In the given \[\triangle\]ABC, right-angled at B and side AB = 24 cm, BC = 7 cm. Find
1. Sin A, Cos A
2. Sin C, Cos C
ABC, right-angled triangle at \[B\] = \[\angle{B}\] = \[90^{\circ}\]
Here it's given: \[AB = 24 cm\] and \[BC = 7 cm\]
That means, \[AC = Hypotenuse\]
According to the Pythagoras Theorem,
The squares of the hypotenuse side are equal to the sum of the squares of the other two sides in a right-angled triangle,
By applying Pythagoras theorem, we get
\[AC^2 = AB^2 + BC^2\]
\[AC^2 = (24)^2 + (7)^2\]
\[AC^2 = (576 + 49)\]
\[AC^2 = 625 cm^2\]
Therefore, \[AC = 25 cm\]
1. Now for Sin A and Cos A.
\[Sin A = \frac{BC}{AC} = \frac{7}{25}\]
Again, the Cosine of an angle is equal to the ratio of the adjacent side and hypotenuse side. Therefore,
\[Cos A = \frac{AB}{AC} = \frac{24}{25}\]
2. For Sin C and Cos C.
\[Sin C = \frac{AB}{AC} = \frac{24}{25}\]
\[Cos C = \frac{BC}{AC} = \frac{7}{25}\]
3. Find the value of the expression \[Sin^{2} 30^{0}\] + \[Cos^{2} 30^{0}\]?
We know that the value of \[Sin^{2} {\theta}\] + \[Cos^{2}{\theta}\] = 1
Therefore \[Sin^{2}30^{\circ}\] + \[Cos^{2}30^{\circ}\] = 1
4. Find the value of \[Cos 105^{0}\].
We can break \[105^{\circ}\] into \[60^{\circ}\] and \[45^{\circ}\] Since those values are relatively easy to find the Cosine of. Therefore the value of \[Cos 105^{\circ} = Cos(60^{\circ} + 45^{\circ})= Cos 60^{\circ} Cos 45^{\circ} - Sin 60^{\circ} Sin 45^{\circ}\] , using the unit circle we obtain, = \[\frac{1}{2} ∙ \frac{1}{\sqrt{2}} - \frac{\sqrt{3}}{2} ∙ \frac{1}{\sqrt{2}} = \frac{1}{2}(\sqrt{2}− \sqrt{6}\]).
5. How do you introduce trigonometry to students initially?
The Vedantu website has expert mathematics teachers to introduce trigonometry and teach them effectively. First, we need to measure the lengths of the sides of sets of similar right-angled triangles and find the ratio of those sides. Then investigate the unique relationship between these ratios and the sizes of the angle. You can make use of your calculators or specific trigonometric tables to find the respective sine, cosine, and tangent of angles. Apply the known ratios of unit triangles, where the value of hypotenuse = 1, to find unknowns in similar triangles.
6. What is the proper origin of trigonometry?
The name trigonometry comes from the Greek words for the triangle as trigon which means three angles and the term metron that means measure. Hellenist mathematicians, around 500-300 BC, experimented by applying trigonometry to the location of stars and other celestial objects in the universe. Therefore, the origins of trigonometry are very well known for practical measurement tasks of finding unknown sides and angles of the right-angled triangles. In the late 1500s, further other advanced trigonometric or circular functions were developed.
7. How is Trigonometry used in real life?
Many real-life situations are evolving around trigonometric functions, such as measuring sea levels every time the tide changes, and measuring the length of each day as seasons change. Trigonometry is also mainly used to set directions such as the north-south or east-west, it will help you decide in what direction you should fix a compass to go in a straight direction. It is used in the navigation of routes in airlines and marine fields in order to pinpoint a location. It is also used to find the appropriate distance of the shore from a point in the sea.
8. How is Pythagoras theorem in trigonometry used?
The Pythagoras theorem which is most famous in the trigonometry subject is useful to measure any two-dimensional navigation. You can use any of the two lengths to find the shortest distance. For instance, just imagine a form of triangle framed in this given example where a plane can actually use its height above the ground level and simultaneously compare its distance from the destination airport to find the correct place to begin the landing action.
9. Does the Pythagoras theorem in trigonometry affect today’s life in any way?
The Pythagoras Theorem has many real-life applications mainly in marine and airways. Now even the rescue workers or firefighters use the Pythagorean Theorem to find out the length of their ladders and their target area, because, just imagine a right-angled triangle formed when the ladder is placed against the wall from the ground, making it possible for the Pythagorean Theorem to be applied effectively. The construction workers and even the astronomers use this theorem to study cosmology and the very universe. Hence now Pythagoras is almost found in every field in some way or the other.
