Trigonometry is the algebra of ratios of the sides of a triangle. The word Trigonometry was derived from greek, with trignos meaning triangle and metry meaning to measure. The earlier study of trigonometry started in the 2nd millennium BC by an Egyptian mathematician by the name Papyrus. Trigonometry finds its applications in other areas of mathematics such as calculus, coordinate geometry, plane geometry, and algebra.
In this mini-lesson, we shall explore the world of trigonometry, by finding answers to questions like what are trigonometric functions, what are complementary angles, what are inverse trigonometric functions, and by understanding the use of these trigonometric ratios by solving examples and interactive questions.
Lesson Plan
Trigonometric Functions
Trigonometry is the algebra of ratios of sides of a right-angled traingle. These ratios are represented as trigonometric ratios of sine(sin), cosine(cos), tangent(tan), cotangent(cot), secant(sec), cosecant(cosec).
The Pythagoras theorem is helpful to build the relationship between the three sides of a right-angled triangle.
\[ \text {Hypotenuse}^2 = \text{Perpendicular}^2 + \text{Base}^2 \]
Further, the ratio of the sides of a right-angled triangle gives the following trigonometric ratios.
| Trigonometric Ratios | Ratio of Sides |
|---|---|
| \( \text{Sin }\theta \) | \( \dfrac{\text{Perpendicular}}{\text {Hypotenuse}} \) |
| \( \text{Cosec }\theta \) | \( \dfrac{\text {Hypotenuse}}{\text{Perpendicular}} \) |
| \( \text{Cos }\theta \) | \( \dfrac{\text{Base}}{\text {Hypotenuse}} \) |
| \( \text{Sec }\theta \) | \( \dfrac{\text {Hypotenuse}}{\text{Base}} \) |
| \( \text{Tan }\theta \) | \( \dfrac{\text{Perpendicular}}{\text{Base}} \) |
| \( \text{Cot }\theta \) | \( \dfrac{\text{Base}}{\text{Perpendicular}} \) |
Complementary angles
The two angles which sum up to a \(90^\circ \) are called complementary angles.
Here the angle \(\theta \) is an acute angle.
The trigonometry ratio formulae for complementary angles is as follows.
- \(\text{Sin }(90^\circ - \theta) = \text{Cos }\theta \) also \(\text{Cos }(90^\circ - \theta) = \text{Sin }\theta \)
- \(\text{Tan }(90^\circ - \theta) = \text{Cot }\theta \) also \(\text{Cot }(90^\circ - \theta) = \text{Tan }\theta \)
- \(\text{Cosec }(90^\circ -\theta) = \text{Sec }\theta\) also \(\text{Sec }(90^\circ - \theta) = \text{Cosec }\theta \)
- Among the trigonometric functions Cos\(\theta\) and Sec\(\theta \) are even functions. \[Cos(-\theta) = Cos \theta\text{ and }Sec(-\theta) = Sec\theta \]
- All the other functions are odd functions. \[ Sin(-\theta) = -Sin \theta ;Tan(-\theta) = -Tan\theta;\\ Cot(-\theta) = -Cot\theta;Cosec(-\theta) = -Cosec\theta \]
- Some of the principal values of Sin \(\theta\), Cos\(\theta\), and Tan\(\theta \) are as follows.
\(\begin{align} Sin0^\circ = 0;Sin30^\circ = \dfrac{1}{2};Sin45^\circ = \dfrac{1}{\sqrt 2};Sin60^\circ = \dfrac{\sqrt 3}{2};Sin90^\circ = 1 \\ Cos0^\circ = 0;Cos30^\circ = \dfrac{\sqrt 3}{2};Cos45^\circ = \dfrac{1}{\sqrt 2};Cos60^\circ = \dfrac{1}{2};Cos90^\circ = 0 \\ \text{Tan }0^\circ = 0;Tan30^\circ = \dfrac{1}{\sqrt 3};Tan45^\circ = 1;Tan60^\circ = \sqrt 3;\\Tan90^\circ = \text{undefined}\end{align} \)
Reciprocal Identities
A set of trigonometric ratios are related through the following reciprocal relation.
- \( \text{Sin }\theta = \dfrac{1}{\text{Cosec }\theta}\) or \( \text{Cosec }\theta = \dfrac{1}{\text{Sin }\theta}\)
- \( \text{Cos }\theta = \dfrac{1}{\text{Sec }\theta}\) or \( \text{Sec }\theta = \dfrac{1}{\text{Cos }\theta}\)
- \( \text{Tan }\theta = \dfrac{1}{\text{Cot }\theta}\) or \( \text{Cot }\theta = \dfrac{1}{\text{Tan }\theta}\)
Also, we have the following important formulae.
- \(\text{Tan }\theta = \dfrac{\text{Sin }\theta}{\text{Cos }\theta}\)
- \(\text{Cot }\theta = \dfrac{\text{Cos }\theta}{\text{Sin }\theta}\)
Pythagorean Identities
The concept of Pythagoras theorem, if applied to trigonometric ratios results in the following three formulae.
- \(\text{Sin }^2\theta + \text{Cos }^2\theta = 1 \)
- \(1 + \text{Tan }^2\theta = \text{Sec }^2\theta \)
- \(1 + \text{Cot }^2\theta = \text{Cosec }^2\theta \)
Inverse Trigonometric Functions
The trigonometric ratios are inverted to create the inverse trigonometric functions.
\[\begin{align} \text{Sin } \theta &= x \\ \theta &= \text{Sin }^{-1} x \end{align} \]
Here x can have values in whole numbers, decimals, fractions, and exponents.
\( \text{Sin }^{-1}(-x) = -\text{Sin }^{-1}(x) \)
\( \text{Cos }^{-1}(-x) =\pi -\text{Cos }^{-1}(x) \)
\( \text{Tan }^{-1}(-x) = -\text{Tan }^{-1}(x) \)
\( \text{Cot }^{-1}(-x) = \pi -\text{Cot }^{-1}(x) \)
\( \text{Sec }^{-1}(-x) = \pi -\text{Sec }^{-1}(x) \)
\( \text{Cosec }^{-1}(-x) = -\text{Cosec }^{-1}(x) \)
Cofunction Identities
The cofunction identities provide the interrelationship between the different inverse trigonometry functions.
- \( \text{Sin }^{-1}(x) = \text{Cosec }^{-1}\dfrac{1}{x} \)
- \( \text{Cos }^{-1}(x) = \text{Sec }^{-1}\dfrac{1}{x} \)
- \( \text{Tan }^{-1}(x) = \text{Cot }^{-1}\dfrac{1}{x} \)
- \( \text{Sin }^{-1}(x) + \text{Cos }^{-1}(x) = \dfrac{\pi}{2} \)
- \( \text{Tan }^{-1}(x) + \text{Cot }^{-1}(x) = \dfrac{\pi}{2} \)
- \( \text{Sec }^{-1}(x) + \text{Cosec }^{-1}(x) = \dfrac{\pi}{2} \)
- \( \text{Sin }^{-1}x = \text{Cos }^{-1}\sqrt{ (1 - x^2) } \)
Each of the trigonometry formulae can be transformed and written in the form of inverse trigonometry formulae. A few of the examples are listed below.
- \( \text{Sin }\theta = \dfrac{1}{\text{Cosec }\theta}\) can be written in inverse trigonometric form as \(\text{Sin }^{-1}x = \text{Cosec }^{-1}\left(\dfrac{1}{x}\right) \)
- \( \text{Sin }\theta = \sqrt {(1 - \text{Cos }^2\theta)}\) can be written in inverse trigonometric form as \(\text{Sin }^{-1}x = \text{Cos }^{-1}\sqrt{(1 - x^2)} \)
- \( \text{Sin }2\theta = 2\text{Sin }\theta.\text{Cos }\theta\) can be written in inverse trigonometric form as 2\(\text{Sin }^{-1}x = \text{Sin }^{-1}2x\sqrt(1 - x^2) \)
- \(\text{Tan}(x + y) = \dfrac{\text{Tan} x + \text{Tan} y}{1 - \text{Tan} x \cdot \text{Tan} y}\) can be written in inverse trigonometric form as \(\text{Tan }^{-1}x + \text{Tan }^{-1}y = \text{Tan }^{-1}\left(\dfrac{x + y}{1 - xy}\right)\)
Now you may apply this idea to transform each of the trigonometric formulae into an inverse trigonometric formula.
Sum and Difference Identities
The combination of two acute angles A and B can be presented through the trigonometric ratios, in the below formulae.
- \(\sin (A + B) = \sin A.\cos B + \cos A.\sin B \)
- \(\sin (A - B) = \sin A.\cos B - \cos A.\sin B \)
- \(\cos (A + B) = \cos A.\cos B -\sin A.\sin B \)
- \(\cos (A - B) = \cos A.\cos B + \sin A.\sin B \)
Double Angle Identities
The double of the angle \(\theta \) is presented through the below few formulae.
- \(\text{Sin }2\theta = 2\text{Sin }\theta.\text{Cos }\theta \)
- \(\begin{align}\text{Cos }2\theta &= 2\text{Cos }^2\theta - 1 \\&= 1 - 2\text{Sin }^2\theta \\&= \text{Cos }^2\theta - \text{Sin }^2\theta\end{align} \)
- \(\text{Tan }2\theta = \dfrac{2\text{Tan }\theta}{1 - \text{Tan }^2\theta} \)
Also we have the \(\text{Sin }2\theta \) and \(\text{Cos } 2\theta \) in terms of \(\text{Tan }\theta \).
- \(\text{Sin }2\theta = \dfrac{2\text{Tan }\theta}{1 +\text{Tan }^2\theta} \)
- \(\text{Cos }2\theta = \dfrac{1 - \text{Tan }^2\theta}{1 +\text{Tan }^2\theta} \)
Half-Angle Identities
The half of the angle \(\theta \) is presented through the below formulae.
- \(\text{Sin }^2\dfrac{\theta }{2} = \dfrac{1 - \text{Cos }\theta}{2} \)
- \(\text{Cos }^2\dfrac{\theta }{2} = \dfrac{1 + \text{Cos }\theta}{2} \)
- \(\text{Sin }^2\dfrac{\theta }{2} + \text{Cos }^2\dfrac{\theta}{2} = 1 \)
Heron's Formula
The Heron's formulae are used to find the area of a triangle. For training of sides of length a, b, c units the Heron's formula to calculate the area is as follows.
A, B, C are the angles of the triangle, and the sides AB = c, BC = a, and AC = b
\[A = \sqrt{(s(s - a)(s - b)(s - c)) }\]
Here, s is the semi perimeter and \(s = \dfrac {a + b + c}{2} \)
Sine Law and Cosine Law
The Sine Law and the Cosine Law gives a relationship between the sides and angles of a triangle.
The Sine Law gives the ratio of the sides and the angle opposite to the side. As an example, the ratio is taken for the side 'a' and its opposite angle 'A'.
\[\dfrac{a}{\text{Sin }A} = \dfrac{b}{\text{Sin }B} = \dfrac{c}{\text{Sin }C} \]
The Cosine Law helps to find the length of a side, for the given lengths of the other two sides and the included angle.
As an example the length 'a' can be found with the help of the other two sides 'b' and 'c' and their included angle 'A'.
\[\begin{align} a^2 &= b^2 + c^2 - 2 bc \text{ Cos }A \\ b^2 &= a^2 + c^2 -2 ac \text{ Cos }B \\ c^2 &= a^2 + b^2 - 2 ab \text{ Cos }C \end{align}\]
a, b,c are the lengths of the sides of the traingle and A, B C are the angles of the traingle.
Solved Examples
| Example 1 |
Raschel is given the trigonometic ratio of \(\text{Tan }\theta = \dfrac{5}{12} \). Help Raschel to find the trigonometric ratio of \(\text{Cosec }\theta \).
Solution
\[\begin{align} \text{Tan }\theta &= \dfrac{\text{Perpendicular}}{\text{Base}} =\dfrac{5}{12} \\ \text{Perpendicular} &= 5 ~ \text{and} ~\text{Base} = 12 \\ \text {Hypotenuse}^2 &= \text{Perpendicular}^2 + \text{Base}^2 \\\text {Hypotenuse}^2 &= 5^2 + 12^2 \\ \text {Hypotenuse}^2 &= 25 + 144 \\ \text {Hypotenuse} &= \sqrt{169}\\ \text {Hypotenuse}&=13\\\text{Hence},~ \text{ Sin }\theta &= \dfrac{\text{Perpendicular}}{\text {Hypotenuse}} = \dfrac{5}{13} ~and~ \text{Cosec }\theta = \dfrac{\text {Hypotenuse}}{\text{Perpendicular}} = \dfrac{13}{5}\end{align}\]
| \(\therefore \text{Cosec }\theta = \dfrac{13}{5} \) |
| Example 2 |
As part of the assignment, Samuel has to find the value of \(\text{Sin }15^\circ \). How can we help Samuel to find the value?
Solution
\[\begin{align} \text{Sin }15^\circ &= \text{Sin }(45^\circ - 30^\circ) \\ &= \text{Sin }45^\circ.\text{Cos }30^\circ - \text{Cos }45^\circ.\text{Sin }30^\circ \\ &= \dfrac{1}{\sqrt2}.\dfrac{\sqrt3}{2} - \dfrac{1}{\sqrt2}.\dfrac{1}{2} \\ &= \dfrac{\sqrt3 - 1}{2\sqrt2} \end{align} \]
| \( \therefore \text{Sin }15^\circ= \dfrac{\sqrt3 - 1}{2\sqrt2} \) |
| Example 3 |
If \(\text{Sin }\theta.\text{Cos }\theta = 5\), find the value of \((\text{Sin }\theta + \text{Cos }\theta)^2\).
Solution
\[\begin{align}(\text{Sin }\theta + \text{Cos }\theta)^2 &= \text{Sin }^2\theta + \text{Cos }^2\theta + 2\text{Sin }\theta.\text{Cos }\theta \\
&= (1) + 2(5) \\&= 1 + 10
\\&=11\end{align}\]
| \(\therefore \) Answer = 11 |
| Example 4 |
If the trigonometric vaue of \(\text{Tan }2A = \text{Cot }(A - 60^\circ)\), find the value of A.
Solution
\[\begin{align}\text{Tan }2A &= \text{Cot }(A - 60^\circ) \\ \text{Cot }(90^\circ - 2A) &= \text{Cot }(A - 60^\circ) \\90^\circ - 2A &= A - 60^\circ \\90^\circ + 60^\circ &= A + 2A \\150^\circ &= 3A \\3A & = 150^\circ \\ A &= \dfrac{150^\circ}{3} \\ A &= 50^\circ\end{align}\]
| \(\therefore~A = 50^\circ \) |
| Example 5 |
Simplify \(\text{Tan }^{-1} \sqrt{\left(\dfrac{1 - \text{Cos }\theta}{1 + \text{Cos }\theta}\right)}\).
Solution
\[\begin{align}\text{Tan }^{-1} \sqrt{\left(\dfrac{1 - \text{Cos }\theta}{1 + \text{Cos }\theta}\right)}&=Tan^{-1}\sqrt{\left(\dfrac{2\text{Sin }^2\dfrac{\theta}{2}}{2\text{Cos }^2\dfrac{\theta}{2}}\right)} \\&=\text{Tan }^{-1} \sqrt{\left(\text{Tan }^2\dfrac{\theta}{2}\right)} \\ &= \text{Tan }^{-1}\text{Tan }\dfrac{\theta}{2} \\ &= \dfrac{\theta}{2}\end{align}\]
| \(\therefore\)The answer is \(\dfrac{\theta}{2} \) |
Interactive Questions on Trigonometry Formulae
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
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FAQs on Trigonometry Formula
1. What is trigonometry?
Trigonometry is the algebra of the trigonometric ratios. There are six trigonometric ratios \(Sin\theta,~ Cos\theta,~ Tan\theta,~ Cot\theta,~ Sec\theta,~and~ Cosec\theta \) which are derived from the ratio of sides of a right-angled triangle.
2. What are trigonometric ratios?
The trigonometric ratios are the ratio of the sides of a right ranged triangle.
\[\begin{align} Sin\theta &= \dfrac{Perp}{Hyp}; ~Cosec\theta = \dfrac{Hyp}{Perp}\\Cos\theta &= \dfrac{Base}{Hyp};~ Sec\theta = \dfrac{Hyp}{Base} \\ Tan\theta &= \dfrac{Perp}{Base}; ~Cot\theta = \dfrac{Base}{Perp} \end{align} \]
3. What is inverse trigonometry ratios?
The Trigonometric ratios are inverted to create inverse trigonometric functions.
\[\begin{align} \text{Sin } \theta &= x \\ \theta &= \text{Sin }^{-1} x \end{align} \]
4. What is the pythagoras theorem?
The Pythagoras theorem gives a relationship between the sides of a right-angled triangle.
\[ (Pythagoras)^2 = (Perpendicular)^2 + (Height)^2\]
5. When do you use Heron's formula?
The Heron's formulae are used to find the area of a triangle. For training of sides of length a, b, c units the Heron's formula to calculate the area is as follows.
\(A = \sqrt(s(s - a)(s - b)(s - c)) \)
Here s is the semi perimeter and \(s = \dfrac {a + b + c}{2} \)
6. What does the cosine rule mean?
The cosine rule helps to find the length of the sides of the traingle, given the length of the other two sides and the included angle..
\[\begin{align} a^2 &= b^2 + c^2 - 2bcCosA \\ b^2 &= a^2 + c^2 -2acCosB \\ c^2 &= a^2 + b^2 - 2anCosC \end{align}\]
7. What does the sine rule state?
The sine rule states the relationship between the sides of the triangle and the sine of the angles of the triangle.
\[\dfrac{a}{SinA} = \dfrac{b}{SinB} = \dfrac{c}{SinC} \]