Trigonometry Formulas

A comprehensive and accessible guide to the most important formulas in trigonometry.

Function Relationships

$\sin\theta = \frac{1}{\csc\theta}$

$\csc\theta = \frac{1}{\sin\theta}$

$\cos\theta = \frac{1}{\sec\theta}$

$\sec\theta = \frac{1}{\cos\theta}$

$\tan\theta = \frac{1}{\cot\theta} = \frac{\sin\theta}{\cos\theta}$

$\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$

Pythagorean Identities

$\sin^2\theta + \cos^2\theta = 1$

$\tan^2\theta + 1 = \sec^2\theta$

$\cot^2\theta + 1 = \csc^2\theta$

Double Angle Formulas

$\sin 2\theta = 2\sin\theta\cos\theta$

$\cos 2\theta = \cos^2\theta - \sin^2\theta$
$= 1 - 2\sin^2\theta$
$= 2\cos^2\theta - 1$

$\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$

Triple Angle Formulas

$\sin 3\theta = 3\sin\theta - 4\sin^3\theta$

$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$

$\tan 3\theta = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$

Opposite Angle Formulas

$\sin(-\theta) = -\sin(\theta)$

$\cos(-\theta) = \cos(\theta)$

$\tan(-\theta) = -\tan(\theta)$

$\cot(-\theta) = -\cot(\theta)$

$\sec(-\theta) = \sec(\theta)$

$\csc(-\theta) = -\csc(\theta)$

Cofunction Formulas (in Quadrant I)

$\sin\theta = \cos(\frac{\pi}{2} - \theta)$

$\cos\theta = \sin(\frac{\pi}{2} - \theta)$

$\tan\theta = \cot(\frac{\pi}{2} - \theta)$

$\cot\theta = \tan(\frac{\pi}{2} - \theta)$

$\sec\theta = \csc(\frac{\pi}{2} - \theta)$

$\csc\theta = \sec(\frac{\pi}{2} - \theta)$

Angle Addition Formulas

$\sin(A+B) = \sin A \cos B + \cos A \sin B$

$\sin(A-B) = \sin A \cos B - \cos A \sin B$

$\cos(A+B) = \cos A \cos B - \sin A \sin B$

$\cos(A-B) = \cos A \cos B + \sin A \sin B$

$\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

$\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

Half Angle Formulas

$\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}}$

$\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}$

$\tan\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$
$= \frac{1-\cos\theta}{\sin\theta}$
$= \frac{\sin\theta}{1+\cos\theta}$

Power Reducing Formulas

$\sin^2\theta = \frac{1-\cos 2\theta}{2}$

$\cos^2\theta = \frac{1+\cos 2\theta}{2}$

$\tan^2\theta = \frac{1-\cos 2\theta}{1+\cos 2\theta}$

Product-to-Sum Formulas

$\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$

$\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]$

$\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$

$\cos A \sin B = \frac{1}{2}[\sin(A+B) - \sin(A-B)]$

Sum-to-Product Formulas

$\sin A + \sin B = 2\sin(\frac{A+B}{2})\cos(\frac{A-B}{2})$

$\sin A - \sin B = 2\sin(\frac{A-B}{2})\cos(\frac{A+B}{2})$

$\cos A + \cos B = 2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2})$

$\cos A - \cos B = -2\sin(\frac{A+B}{2})\sin(\frac{A-B}{2})$

Arc Length

$S = r\theta$

Law of Sines

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Law of Tangents

$\frac{a-b}{a+b} = \frac{\tan[\frac{1}{2}(A-B)]}{\tan[\frac{1}{2}(A+B)]}$

Law of Cosines

$a^2 = b^2 + c^2 - 2bc \cos A$

$b^2 = a^2 + c^2 - 2ac \cos B$

$c^2 = a^2 + b^2 - 2ab \cos C$

25 Solved Quadratic Equations with Step-by-Step Solutions

Mastering Quadratic Equations

What you will learn

A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. The standard form is $ ax^2 + bx + c = 0 $. Below are 25 fully solved examples to help you understand factorization and formula methods, followed by 10 practice problems to test your skills.

Note: Try to solve the equation yourself first, then click "Show Solution" to verify your method.

25 Solved Examples

01 Solve: $ x^2 + 5x + 6 = 0 $

View Step-by-Step Solution

(i) Identify coefficients Here, $a=1, b=5, c=6$. We need two numbers that multiply to 6 and add to 5.

(ii) Factorize The numbers are 2 and 3. \[ x^2 + 2x + 3x + 6 = 0 \] \[ x(x + 2) + 3(x + 2) = 0 \] \[ (x + 2)(x + 3) = 0 \]
(iii) Find Roots Either $x+2=0$ or $x+3=0$.
Answer: $ x = -2, x = -3 $

02 Solve: $ x^2 - 5x + 6 = 0 $

View Step-by-Step Solution

(i) Find factors We need numbers that multiply to $+6$ and add to $-5$. These are $-2$ and $-3$.

(ii) Split the middle term \[ x^2 - 2x - 3x + 6 = 0 \] \[ x(x - 2) - 3(x - 2) = 0 \] \[ (x - 2)(x - 3) = 0 \]
(iii) Solve Answer: $ x = 2, x = 3 $

03 Solve: $ x^2 - 9 = 0 $

View Step-by-Step Solution

(i) Use identity Recall $ a^2 - b^2 = (a+b)(a-b) $. Here $ 9 = 3^2 $.

(ii) Factorize \[ x^2 - 3^2 = 0 \] \[ (x + 3)(x - 3) = 0 \]
(iii) Solve Answer: $ x = -3, x = 3 $

04 Solve: $ x^2 + 7x + 12 = 0 $

View Step-by-Step Solution

(i) Find factors Product = 12, Sum = 7. Factors are 3 and 4.

(ii) Factorize \[ (x + 3)(x + 4) = 0 \]
(iii) Solve Answer: $ x = -3, x = -4 $

05 Solve: $ 2x^2 + 3x + 1 = 0 $

View Step-by-Step Solution

(i) AC Method Multiply $a \times c = 2 \times 1 = 2$. We need sum = 3. Factors are 2 and 1.

(ii) Split middle term \[ 2x^2 + 2x + x + 1 = 0 \] \[ 2x(x + 1) + 1(x + 1) = 0 \] \[ (2x + 1)(x + 1) = 0 \]
(iii) Solve $ 2x = -1 \rightarrow x = -1/2 $ or $ x = -1 $.
Answer: $ x = -\frac{1}{2}, x = -1 $

06 Solve: $ x^2 - 2x - 15 = 0 $

View Step-by-Step Solution

(i) Factors Product = -15, Sum = -2. Factors: -5 and +3.

(ii) Factorize \[ (x - 5)(x + 3) = 0 \]
(iii) Solve Answer: $ x = 5, x = -3 $

07 Solve: $ x^2 - 8x + 16 = 0 $

View Step-by-Step Solution

(i) Analyze structure This is a perfect square trinomial because $ (-4)^2 = 16 $ and $ 2(-4) = -8 $.

(ii) Factorize \[ (x - 4)^2 = 0 \]
(iii) Solve Answer: $ x = 4 $ (Equal real roots)

08 Solve: $ 3x^2 - 5x + 2 = 0 $

View Step-by-Step Solution

(i) AC Method $ a \times c = 3 \times 2 = 6 $. Sum = -5. Factors: -3 and -2.

(ii) Split middle term \[ 3x^2 - 3x - 2x + 2 = 0 \] \[ 3x(x - 1) - 2(x - 1) = 0 \] \[ (3x - 2)(x - 1) = 0 \]
(iii) Solve Answer: $ x = \frac{2}{3}, x = 1 $

09 Solve: $ x^2 + 4x = 0 $

View Step-by-Step Solution

(i) Take common factor There is no constant term $c$. Take $x$ common.

(ii) Factorize \[ x(x + 4) = 0 \]
(iii) Solve Answer: $ x = 0, x = -4 $

10 Solve: $ 2x^2 - 7x + 3 = 0 $

View Step-by-Step Solution

(i) AC Method Product = 6, Sum = -7. Factors: -6 and -1.

(ii) Split middle term \[ 2x^2 - 6x - x + 3 = 0 \] \[ 2x(x - 3) - 1(x - 3) = 0 \] \[ (2x - 1)(x - 3) = 0 \]
(iii) Solve Answer: $ x = \frac{1}{2}, x = 3 $

11 Solve: $ x^2 - 4x - 21 = 0 $

View Step-by-Step Solution

(i) Factors Product = -21, Sum = -4. Factors: -7 and +3.

(ii) Factorize \[ (x - 7)(x + 3) = 0 \]
(iii) Solve Answer: $ x = 7, x = -3 $

12 Solve using Formula: $ x^2 + 4x + 2 = 0 $

View Step-by-Step Solution

(i) Quadratic Formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ where $a=1, b=4, c=2$.

(ii) Substitute \[ x = \frac{-4 \pm \sqrt{16 - 8}}{2} \] \[ x = \frac{-4 \pm \sqrt{8}}{2} \] \[ x = \frac{-4 \pm 2\sqrt{2}}{2} \]
(iii) Simplify Answer: $ x = -2 \pm \sqrt{2} $

13 Solve: $ 6x^2 - x - 2 = 0 $

View Step-by-Step Solution

(i) AC Method Product = -12, Sum = -1. Factors: -4 and +3.

(ii) Split middle term \[ 6x^2 - 4x + 3x - 2 = 0 \] \[ 2x(3x - 2) + 1(3x - 2) = 0 \] \[ (2x + 1)(3x - 2) = 0 \]
(iii) Solve Answer: $ x = -\frac{1}{2}, x = \frac{2}{3} $

14 Solve: $ 4x^2 - 12x + 9 = 0 $

View Step-by-Step Solution

(i) Identify square $ (2x)^2 - 2(2x)(3) + 3^2 = 0 $.

(ii) Factorize \[ (2x - 3)^2 = 0 \]
(iii) Solve Answer: $ x = \frac{3}{2} $ (Repeated root)

15 Solve: $ x^2 + x - 2 = 0 $

View Step-by-Step Solution

(i) Factors Product = -2, Sum = +1. Factors: +2 and -1.

(ii) Factorize \[ (x + 2)(x - 1) = 0 \]
(iii) Solve Answer: $ x = -2, x = 1 $

16 Solve: $ 5x^2 = 20 $

View Step-by-Step Solution

(i) Isolate x squared Divide by 5: $ x^2 = 4 $.

(ii) Square root $ x = \pm\sqrt{4} $.
(iii) Solve Answer: $ x = 2, x = -2 $

17 Solve: $ x^2 - 11x + 24 = 0 $

View Step-by-Step Solution

(i) Factors Product = 24, Sum = -11. Factors: -8 and -3.

(ii) Factorize \[ (x - 8)(x - 3) = 0 \]
(iii) Solve Answer: $ x = 8, x = 3 $

18 Solve: $ x^2 + 10x + 25 = 0 $

View Step-by-Step Solution

(i) Perfect Square This fits $ (a+b)^2 $.

(ii) Factorize \[ (x + 5)^2 = 0 \]
(iii) Solve Answer: $ x = -5 $

19 Solve: $ 2x^2 + 5x - 3 = 0 $

View Step-by-Step Solution

(i) AC Method Product = -6, Sum = 5. Factors: +6 and -1.

(ii) Split middle term \[ 2x^2 + 6x - x - 3 = 0 \] \[ 2x(x + 3) - 1(x + 3) = 0 \] \[ (2x - 1)(x + 3) = 0 \]
(iii) Solve Answer: $ x = \frac{1}{2}, x = -3 $

20 Solve using Formula: $ x^2 - 6x + 7 = 0 $

View Step-by-Step Solution

(i) Apply Formula $ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(7)}}{2(1)} $.

(ii) Calculate \[ x = \frac{6 \pm \sqrt{36 - 28}}{2} \] \[ x = \frac{6 \pm \sqrt{8}}{2} \] \[ x = \frac{6 \pm 2\sqrt{2}}{2} \]
(iii) Simplify Answer: $ x = 3 \pm \sqrt{2} $

21 Solve: $ x^2 + 3x - 10 = 0 $

View Step-by-Step Solution

(i) Factors Product = -10, Sum = 3. Factors: +5 and -2.

(ii) Factorize \[ (x + 5)(x - 2) = 0 \]
(iii) Solve Answer: $ x = -5, x = 2 $

22 Solve: $ 3x^2 = 2x $

View Step-by-Step Solution

(i) Rearrange \[ 3x^2 - 2x = 0 \]
(ii) Factorize common term \[ x(3x - 2) = 0 \]
(iii) Solve $ x = 0 $ or $ 3x - 2 = 0 $.
Answer: $ x = 0, x = \frac{2}{3} $

23 Solve: $ x^2 - 13x + 42 = 0 $

View Step-by-Step Solution

(i) Factors Product = 42, Sum = -13. Factors: -6 and -7.

(ii) Factorize \[ (x - 6)(x - 7) = 0 \]
(iii) Solve Answer: $ x = 6, x = 7 $

24 Solve: $ 4x^2 - 1 = 0 $

View Step-by-Step Solution

(i) Difference of Squares $ (2x)^2 - 1^2 = 0 $.

(ii) Factorize \[ (2x - 1)(2x + 1) = 0 \]
(iii) Solve Answer: $ x = \frac{1}{2}, x = -\frac{1}{2} $

25 Solve: $ x^2 + 8x + 15 = 0 $

View Step-by-Step Solution

(i) Factors Product = 15, Sum = 8. Factors: 3 and 5.

(ii) Factorize \[ (x + 3)(x + 5) = 0 \]
(iii) Solve Answer: $ x = -3, x = -5 $

Practice Questions

Try solving these 10 questions on your own before checking the answer key below.

(i) $ x^2 + 7x + 10 = 0 $

(ii) $ x^2 - 3x - 10 = 0 $

(iii) $ 2x^2 + 5x + 3 = 0 $

(iv) $ x^2 - 49 = 0 $

(v) $ x^2 - 6x = 0 $

(vi) $ x^2 + 12x + 36 = 0 $

(vii) $ 3x^2 - x - 4 = 0 $

(viii) $ x^2 - x - 30 = 0 $

(ix) $ x^2 - 10x + 21 = 0 $

(x) $ 2x^2 + 7x - 4 = 0 $

Check Answer Key

(i) $ x = -2, x = -5 $

(ii) $ x = 5, x = -2 $

(iii) $ x = -1, x = -3/2 $

(iv) $ x = 7, x = -7 $

(v) $ x = 0, x = 6 $

(vi) $ x = -6 $

(vii) $ x = -1, x = 4/3 $

(viii) $ x = 6, x = -5 $

(ix) $ x = 3, x = 7 $

(x) $ x = 1/2, x = -4 $

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