Math Formula SPM
SPM Math Formulas: Complete Guide with Examples
SPM (Sijil Pelajaran Malaysia) Mathematics is a crucial subject for students, covering a wide range of topics such as algebra, geometry, trigonometry, and calculus. Mastering the key formulas is essential for success in the exam. This article provides a comprehensive list of essential SPM math formulas along with examples to help students understand and apply them effectively.
1. Algebra Formulas
Quadratic Formula
The quadratic formula is used to solve quadratic equations of the form ax2 + bx + c = 0. The formula is given by:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Example: Solve the equation 2x2 + 5x - 3 = 0.
Solution:
Here, a = 2, b = 5, and c = -3. Applying the quadratic formula:
$$x = \frac{-5 \pm \sqrt{5^2 - 4(2)(-3)}}{2(2)} = \frac{-5 \pm \sqrt{25 + 24}}{4} = \frac{-5 \pm \sqrt{49}}{4}$$
Therefore, the solutions are x = 1 and x = -1.5.
Discriminant of a Quadratic Equation
The discriminant D of a quadratic equation is given by:
$$D = b^2 - 4ac$$
The discriminant helps determine the nature of the roots of the equation:
- If D > 0, the equation has two distinct real roots.
- If D = 0, the equation has one repeated real root.
- If D < 0, the equation has two complex roots.
Arithmetic Progression (AP)
An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. The formula for the n-th term is:
$$T_n = a + (n - 1)d$$
The sum of the first n terms is:
$$S_n = \frac{n}{2}(2a + (n - 1)d)$$
Example: Find the 10th term of the AP where the first term a = 3 and the common difference d = 2.
Solution:
$$T_{10} = 3 + (10 - 1) \times 2 = 3 + 18 = 21$$
Geometric Progression (GP)
A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio r. The formula for the n-th term is:
$$T_n = ar^{n-1}$$
The sum of the first n terms of a geometric progression is:
$$S_n = a \frac{1 - r^n}{1 - r} \quad \text{(if } r \neq 1\text{)}$$
Example: Find the 5th term of a GP where the first term a = 2 and the common ratio r = 3.
Solution:
$$T_5 = 2 \times 3^{5-1} = 2 \times 81 = 162$$
2. Geometry Formulas
Area of a Triangle
The area of a triangle can be calculated using the formula:
$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$
Example: Find the area of a triangle with base 8 cm and height 5 cm.
Solution:
$$\text{Area} = \frac{1}{2} \times 8 \times 5 = 20 \text{ cm}^2$$
Area of a Circle
The area of a circle is given by:
$$\text{Area} = \pi r^2$$
where r is the radius of the circle.
Example: Find the area of a circle with radius 7 cm.
Solution:
$$\text{Area} = \pi \times 7^2 = 49\pi \approx 153.94 \text{ cm}^2$$
Pythagoras Theorem
Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The formula is:
$$c^2 = a^2 + b^2$$
Example: In a right-angled triangle, if a = 3 cm and b = 4 cm, find the hypotenuse c.
Solution:
$$c^2 = 3^2 + 4^2 = 9 + 16 = 25 \Rightarrow c = \sqrt{25} = 5 \text{ cm}$$
3. Trigonometry Formulas
Basic Trigonometric Ratios
The basic trigonometric ratios are defined as follows for a right-angled triangle:
- $$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$
- $$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$
- $$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$$
Example: If \theta is an angle in a right-angled triangle where the opposite side is 4 cm and the hypotenuse is 5 cm, find \sin \theta.
Solution:
$$\sin \theta = \frac{4}{5}$$
Trigonometric Identities
Some common trigonometric identities are:
- $$\sin^2 \theta + \cos^2 \theta = 1$$
- $$1 + \tan^2 \theta = \sec^2 \theta$$
- $$1 + \cot^2 \theta = \csc^2 \theta$$
Area of a Sector
The area of a sector of a circle, with a central angle \theta (in radians) and radius r, is given by the formula:
$$\text{Area} = \frac{1}{2} r^2 \theta$$
If the angle \theta is given in degrees, the formula becomes:
$$\text{Area} = \frac{\theta}{360} \times \pi r^2$$
Example: Find the area of a sector with a radius of 10 cm and a central angle of 45 degrees.
Solution:
Using the formula for the area of a sector:
$$\text{Area} = \frac{45}{360} \times \pi \times 10^2 = \frac{1}{8} \times \pi \times 100 = 12.5\pi \approx 39.27 \text{ cm}^2$$
Conclusion
Mastering SPM math formulas is essential for students aiming to excel in their exams. This guide covered key formulas in algebra, geometry, and trigonometry, along with detailed examples to aid understanding. Regular practice and familiarization with these formulas will help students tackle SPM math problems with confidence. Remember, consistent effort and problem-solving practice are the keys to success. Good luck with your SPM preparation!
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