Introduction
Trigonometry deals with the relationships between the angles and sides of triangles. In this chapter, you will learn about trigonometric ratios (sin, cos, tan, cosec, sec, cot) for acute angles of a right triangle. You will learn the specific values of these ratios for standard angles (0, 30, 45, 60, 90 degrees) and explore important trigonometric identities that are used extensively in higher mathematics.
Trigonometric Ratios
In a right-angled triangle, the trigonometric ratios are defined for each acute angle in terms of the sides of the triangle. For an angle theta: sin(theta) = opposite/hypotenuse, cos(theta) = adjacent/hypotenuse, tan(theta) = opposite/adjacent. The reciprocal ratios are: cosec(theta) = 1/sin(theta), sec(theta) = 1/cos(theta), cot(theta) = 1/tan(theta). Also, tan(theta) = sin(theta)/cos(theta) and cot(theta) = cos(theta)/sin(theta).
Key Points
- •sin = Opposite/Hypotenuse (SOH)
- •cos = Adjacent/Hypotenuse (CAH)
- •tan = Opposite/Adjacent (TOA)
- •cosec = 1/sin, sec = 1/cos, cot = 1/tan
- •tan = sin/cos and cot = cos/sin
Worked Example
In a right triangle with angle A, if the side opposite to A is 3 and hypotenuse is 5, find all trigonometric ratios. Adjacent = sqrt(5^2 - 3^2) = sqrt(16) = 4 sin A = 3/5, cos A = 4/5, tan A = 3/4 cosec A = 5/3, sec A = 5/4, cot A = 4/3
Watch Out
Use the mnemonic SOH-CAH-TOA to remember the basic ratios. 'Some Old Horses Can Always Hear Their Owner Approaching.'
Trigonometric Ratios of Standard Angles
The trigonometric ratios of certain standard angles (0, 30, 45, 60, 90 degrees) can be calculated exactly. These values must be memorised as they appear in almost every trigonometry problem. sin: 0, 1/2, 1/sqrt(2), sqrt(3)/2, 1 cos: 1, sqrt(3)/2, 1/sqrt(2), 1/2, 0 tan: 0, 1/sqrt(3), 1, sqrt(3), undefined These values follow a pattern and are derived from the properties of equilateral and isosceles right triangles.
Key Points
- •sin 0 = 0, sin 30 = 1/2, sin 45 = 1/sqrt(2), sin 60 = sqrt(3)/2, sin 90 = 1
- •cos values are sin values in reverse order
- •tan 0 = 0, tan 30 = 1/sqrt(3), tan 45 = 1, tan 60 = sqrt(3), tan 90 = undefined
- •sin and cos are always between 0 and 1 (for acute angles)
- •tan and cot can be any non-negative value for acute angles
Worked Example
Evaluate: sin 60.cos 30 + sin 30.cos 60 = (sqrt(3)/2)(sqrt(3)/2) + (1/2)(1/2) = 3/4 + 1/4 = 1 Note: This equals sin(60 + 30) = sin 90 = 1
Watch Out
Learn the trick: for sin, write 0,1,2,3,4 under the angles 0,30,45,60,90. Divide by 4 and take the square root. sin 30 = sqrt(1/4) = 1/2.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric ratios that are true for all values of the angle. The three fundamental identities are: (1) sin^2(theta) + cos^2(theta) = 1, (2) 1 + tan^2(theta) = sec^2(theta), (3) 1 + cot^2(theta) = cosec^2(theta). These identities are derived from the Pythagoras theorem and are used to simplify trigonometric expressions and prove other identities.
Key Points
- •sin^2(A) + cos^2(A) = 1
- •1 + tan^2(A) = sec^2(A)
- •1 + cot^2(A) = cosec^2(A)
- •These can be rearranged: sin^2(A) = 1 - cos^2(A), etc.
- •Used to prove identities: usually work on LHS to reach RHS (or vice versa)
Worked Example
Prove that (1 - cos^2 A) cosec^2 A = 1 LHS = (1 - cos^2 A) cosec^2 A = sin^2 A . cosec^2 A [using sin^2 A + cos^2 A = 1] = sin^2 A . (1/sin^2 A) = 1 = RHS. Hence proved.
Watch Out
When proving identities, convert everything to sin and cos first. This makes simplification much easier.
Quick Summary
- ✓Six trigonometric ratios: sin, cos, tan, cosec, sec, cot
- ✓SOH-CAH-TOA for remembering definitions
- ✓Standard angle values must be memorised for 0, 30, 45, 60, 90 degrees
- ✓sin^2(A) + cos^2(A) = 1 (most fundamental identity)
- ✓1 + tan^2(A) = sec^2(A)
- ✓1 + cot^2(A) = cosec^2(A)
- ✓To prove identities: convert to sin/cos, use algebraic techniques
Key Formulas
sin = Opposite/Hypotenuse, cos = Adjacent/Hypotenuse, tan = Opposite/Adjacent
sin^2(A) + cos^2(A) = 1
1 + tan^2(A) = sec^2(A)
1 + cot^2(A) = cosec^2(A)
sin 30 = 1/2, sin 45 = 1/sqrt(2), sin 60 = sqrt(3)/2
cos 30 = sqrt(3)/2, cos 45 = 1/sqrt(2), cos 60 = 1/2
tan 30 = 1/sqrt(3), tan 45 = 1, tan 60 = sqrt(3)
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