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Revision Notes: Trigonometry | Mathematics Class 9 ICSE PDF Download
- Trigonometry is the study of relationship between the sides and the angles of the triangle.
- The word trigonometry is derived from the Greek words ‘tri’ meaning three, ‘gon’ meaning sides and ‘metron’ meaning measure.
- Angle measured in anticlockwise direction is taken as positive angle.
- Angle measured in clockwise direction is taken as negative angle.
- Ratio of the sides of the right triangle with respect to the acute angles is called trigonometric ratios of the angle.
- Trigonometric ratios of acute angle A in right triangle ABC:
- Each trigonometric ratio is a real number. It has no unit.
- Only symbols cosine, sine, tangent, cotangent, sec and cosec have no meaning.
- (sinθ)n is generally written as sinn θ, n being a positive integer. Similarly, other trigonometric ratios can also be written.
- The values of the trigonometric ratios of an angle do not vary with the length of the sides of the triangle, if the angles remain the same.
- Pythagoras theorem: In a right triangle, square of the hypotenuse is equal to the sum of the square of the other two sides.
- When any two sides of a right triangle are given, its third side can be obtained by using Pythagoras theorem.
- Relation between trigonometric ratios:
(i) tanθ = sinθ/cosθ
(ii) cosecθ = 1/sinθ
(iii) secθ = 1/cosθ
(iv) cotθ =
- Values of Trigonometric ratios of some specific angles:
- The value of sin A or cos A never exceeds 1, whereas the value of sec A or cosec A is always greater than 1 or equal to 1.
- The value of sinθ increases from 0 to 1 when θ increases from 0° to 90°.
- The value of cosθ decreases from 1 to 0 when θ increases from 0° to 90°.
- Trigonometric ratios of complementary angles:
(i) sin (90° – A) = cos A
(ii) cos (90° – A) = sin A
(iii) tan (90° – A) = cot A
(iv) cot (90° – A) = tan A
(v) sec (90° – A) = cosec A
(vi) cosec (90° – A) = sec A - An equation involving trigonometric ratios of an angle, say θ, is termed as a trigonometric identity if it is satisfied by all values of θ.
- Basic trigonometric identities:
(i) sin2θ + cos2θ = 1
(ii) 1 + tan2θ = sec2θ
(iii) 1 + cot2θ = sec2θ
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FAQs on Revision Notes: Trigonometry - Mathematics Class 9 ICSE
| 1. What is the basic definition of trigonometry? |  |
Ans. Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right-angled triangles. It involves the use of various functions like sine, cosine, and tangent to relate these angles and sides.
| 2. What are the main trigonometric ratios? | |
Ans. The main trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). In a right-angled triangle, these ratios are defined as follows:
- Sine is the ratio of the length of the opposite side to the length of the hypotenuse (sin = opposite/hypotenuse).
- Cosine is the ratio of the length of the adjacent side to the length of the hypotenuse (cos = adjacent/hypotenuse).
- Tangent is the ratio of the length of the opposite side to the length of the adjacent side (tan = opposite/adjacent).
| 3. How can I remember the trigonometric ratios? | |
Ans. A common mnemonic to remember the trigonometric ratios is "SOH-CAH-TOA." This stands for:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent.
Using this mnemonic can help students recall the definitions of the sine, cosine, and tangent functions easily.
| 4. What is the Pythagorean theorem and how is it related to trigonometry? | |
Ans. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be expressed as a² + b² = c². In trigonometry, this theorem is used to derive the relationships between the trigonometric ratios and to solve various problems involving right triangles.
| 5. How do I solve basic trigonometric equations? | |
Ans. To solve basic trigonometric equations, you can follow these steps:
1. Isolate the trigonometric function on one side of the equation.
2. Use known values of trigonometric ratios or identities to simplify the equation.
3. Find the general solutions using the unit circle or reference angles if necessary.
4. Finally, check for any restrictions based on the domain of the trigonometric function.
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