Trigonometric Identities: Formula of trigonometry class 12 involves trigonometric functions and trigonometric identities. These identities of trigonometry are accurate for all variable’s values.

Trigonometric Ratio: The relationship of the angle measurement and the right-angle triangle side length is known for its trigonometric ratio.

Sum And Difference Of Angles:

Sin A Sin B = ½ [cos(A-B)-cos(A+B)]

cos A cos B = ½ [cos(A-B)+ cos(A+B)]

Sin A cos B = ½ [sin(A+B)+ sin(A-B)]

cos A sin B = ½ [cos(A+B)+ sin(A-B)]

Double Angle Formulas:

sin2θ= 2sinθcosθ

cos2θ= Cos²θ - Sin²θ

= 2Cos²θ - 1

= 1-2Sin²θ

tan2θ= \[\frac{2Tanθ}{1-Tan^{2}θ}\]

Triple Angle Formula:

sin3θ= 3Sinθ - 4\[Sin^{3}θ\]

cos3θ= 4\[Cos^{3}θ\] - 3Cosθ

tan3θ= (3tanθ-tan³θ) / (1-3tan²θ)

cot3θ= \[\frac{Cot^{3}θ - 3Cotθ}{3Cot^{2}θ - 1}\]

Converting Products Into Sums And Difference:

sinA sinB = ½ [cos(A-B) - cos(A+B)]

cosA cosB = ½ [cos(A-B)+cos(A+B)]

sinA cosB = ½ [sin(A+B)+sin(A-B)]

cosA sinB = ½ [cos(A+B)+sin(A-B)]

Half Angle Identities:

Sin(\[\frac{x}{2}\]) = ±\[\sqrt{\frac{1-2cos(x)}{2}}\]

Tan(\[\frac{x}{2}\]) = ±\[\sqrt{\frac{1-cos(x)}{1+Cos(x)}}\] = \[\frac{1-Cos(x)}{Sin(x)}\] = \[\frac{Sin(x)}{1+Cos(x)}\]

Cos(\[\frac{x}{2}\]) = ±\[\sqrt{\frac{1+cos(x)}{2}}\]

On squaring the above identities we can re-state the above equations as following-

\[Sin^{2}(x)\] = \[\frac{1}{2}\][1-Cos(2x)]

\[Cos^{2}(x)\] = \[\frac{1}{2}\][1+Cos(2x)]

\[Tan^{2}(x)\] = \[\frac{1-Cos(2x)}{1+Cos(2x)}\]

Complex Relations:

Sinθ = \[\frac{e^{iθ} - e^{-iθ}}{2i}\]

Cosθ = \[\frac{e^{iθ} + e^{-iθ}}{2i}\]

Tanθ = \[\frac{e^{iθ} - e^{-iθ}}{(e^{iθ} + e^{-iθ})i}\]

Cosecθ = \[\frac{2i}{e^{iθ} - e^{-iθ}}\]

Secθ =\[\frac{2i}{e^{iθ} - e^{-iθ}}\]

Cotθ =\[\frac{(e^{iθ} + e^{-iθ})i}{e^{iθ} - e^{-iθ}}\]

Definition:

θ = \[Sin^{−1}(x)\] is equivalent to x = Sin θ

θ = \[Cos^{−1}(x)\] is equivalent to x = Cos θ

θ = \[Tan^{−1}(x)\] is equivalent to x = Tan θ

Inverse Properties:

These properties hold for x in the domain and θ in the range

sin(sin^{−1} (x)) = x

cos(cos^{−1} (x)) = x

tan(tan^{−1} (x)) = x

sin^{−1} (sin(θ)) = θ

cos^{−1} (cos(θ)) = θ

tan^{−1} (tan(θ)) = θ

Other Notations

sin^{−1} (x) = arcsin(x)

cos^{−1} (x) = arccos(x)

tan^{−1} (x) = arctan(x)

Question 1) Find the principal values of \[Sin^{-1}\] (\[\frac{1}{\sqrt{2}}\])

Solution) Let \[Sin^{-1}\](\[\frac{1}{\sqrt{2}}\]) = α; then sin α =\[\frac{1}{\sqrt{2}}\]= sin 45 degree

α = 45 degree or \[\frac{π}{4}\], which is the required principal value.

Question 2) Show that \[Tan^{-1}\]\[\frac{1}{2}\] + \[Tan^{-1}\]\[\frac{1}{3}\] = \[\frac{π}{4}\]

Solution) L.H.S. = \[Tan^{-1}\] \[\frac{1}{2}\] + \[Tan^{-1}\] \[\frac{1}{3}\] = \[Tan^{-1}\] = \[\frac{\frac{1}{2} + \frac{1}{3}}{1- \frac{1}{2} X \frac{1}{3}}\] = \[Tan^{-1}\] (\[\frac{\frac{5}{6}}{\frac{5}{6}}\]) = \[Tan^{-1}\] = \[\frac{π}{4}\]

The word "Trigonometry" is taken from the word "Triangle Measure".

Trigonometry is used by the engineers to figure out the angles of the sound waves and how to design rooms.

Trigonometry is connected with music and architecture.

FAQ (Frequently Asked Questions)

1. What does quotient relations mean?

In trigonometry class 12, we come across quotient relations. Quotient relations comprise three trigonometric ratios. One is the quotient that we get after division operation between the other two. For example, tan θ = sin θ /cos θ and cot θ = cos θ / sin θ.

2. Who is the father of trigonometry?

Trigonometry is the study between the relationships dealing with angles, heights and lengths of triangles and also the relationships between the different circle parts and other geometric figures. Hipparchus, a Greek astronomer, is considered as the father of trigonometry. 12th Trigonometry formulas contain some trigonometry and its tables were developed by him. Hipparchus was the first person whose quantitative and accurate models for the motion of the sun and moon survive. Also, the first accurate star map is formulated by him. He may have been the first to develop a reliable method to predict a solar eclipse with the help of his solar and lunar theories.

3. What are the laws of trigonometry?

Trigonometry class 12 deals with trigonometric laws of sines and cosines which is helpful in solving a triangle i.e., finding all three angles and sides of any triangle given three out of the six. The law of sines states that in any triangle, not just a right angle triangle but the ratio of the sine of each angle to its opposing side is also equivalent for all three angles. The law of cosines states that in any triangle, the sum of the squares of the other two sides is equal to the square of one side. The trigonometric law of cosines is helpful in solving a triangle if all three sides, or two sides and the included angle are given.