01 · Explore
Understanding Angles and Rotation
An angle is a measure of rotation of a given ray about its initial point.
The original ray is called the initial side, and the final position after rotation is the terminal side. The point of rotation is known as the vertex. The direction of rotation determines the sign of the angle: anticlockwise rotation results in a positive angle, while clockwise rotation results in a negative angle.
The measure of an angle is the amount of rotation performed to get the terminal side from the initial side. While a complete revolution is a natural unit, we primarily use degree and radian measures for mathematical precision.
Components of an Angle
- 1
Initial Side
The starting position of the ray before rotation.
- 2
Vertex
The fixed point about which the ray rotates.
- 3
Rotation
The movement (clockwise or anticlockwise) to the final position.
- 4
Terminal Side
The final position of the ray after rotation is complete.
The process of forming an angle through rotation from an initial to a terminal position.
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Question
What determines if an angle is positive or negative?
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02 · Explore
Degree and Radian Measures
Angles are measured using two primary systems: the sexagesimal system (degrees) and the circular system (radians).
In the degree measure, one complete revolution is divided into 360 equal parts, each called a degree (1°). Degrees are further subdivided: 1 degree equals 60 minutes (1° = 60′), and 1 minute equals 60 seconds (1′ = 60″).
The radian measure is defined based on a unit circle (radius = 1). An angle subtended at the centre by an arc of length 1 unit is 1 radian. More generally, in a circle of radius r, an arc of length l subtends an angle θ = l/r radians. Thus, l = rθ.
| Unit | Symbol | Subdivision 1 | Subdivision 2 |
|---|---|---|---|
| Degree | ° | 1° = 60 minutes | 1 minute = 60 seconds |
| Radian | rad | θ = l / r | Full circle = 2π rad |
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Define 1 radian.
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03 · Explore
Relation Between Degree and Radian
Since a full circle is 360° and also 2π radians, we can establish a direct conversion factor.
The fundamental relation is π radians = 180°. From this, we derive that 1 radian is approximately 57° 16′ and 1° is approximately 0.01746 radians. When an angle is expressed without units (e.g., θ = π/2), it is understood to be in radians.
To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.
Converting Degrees to Radians
40° 20′ = 40 + (20/60) = 121/3 degrees
Convert minutes to degrees first: 20′ = 1/3°. Total degrees = 121/3. Multiply by π/180: (121/3) * (π/180) = 121π/540 radians.
Converting Radians to Degrees
6 radians = 6 * (180/π)
Using π = 22/7, 6 * 180 * 7 / 22 = 3780/11 degrees. This equals 343° and 7/11 of a degree. 7/11 * 60 = 38.18 minutes. 0.18 * 60 ≈ 11 seconds. Result: 343° 38′ 11″.
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Convert 90° to radians.
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04 · Explore
Trigonometric Functions and the Unit Circle
Trigonometric ratios are generalized to functions using a unit circle in the coordinate plane.
Consider a unit circle with centre at the origin. Let P(a, b) be a point on the circle such that the angle AOP is x radians. We define cos x = a and sin x = b. Since the point lies on the unit circle, a² + b² = 1, which gives the fundamental identity: cos² x + sin² x = 1.
Other functions are defined as: cosec x = 1/sin x, sec x = 1/cos x, tan x = sin x/cos x, and cot x = cos x/sin x. These are defined only where the denominators are non-zero.
| Function | 0 | π/6 (30°) | π/4 (45°) | π/3 (60°) | π/2 (90°) |
|---|---|---|---|---|---|
| sin x | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos x | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan x | 0 | 1/√3 | 1 | √3 | Not Defined |
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For what values of x is tan x not defined?
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05 · Explore
Signs of Trigonometric Functions
The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies.
In the first quadrant (I), all trigonometric functions are positive. In the second quadrant (II), only sin and cosec are positive. In the third quadrant (III), only tan and cot are positive. In the fourth quadrant (IV), only cos and sec are positive.
This can be remembered by the mnemonic 'All Silver Tea Cups' or 'Add Sugar To Coffee'. Additionally, sin(-x) = -sin x (odd function) and cos(-x) = cos x (even function).
Finding Functions in Quadrants
cos x = -3/5, x in Quadrant III
In Quadrant III, sin x is negative. sin² x = 1 - cos² x = 1 - 9/25 = 16/25. So, sin x = -4/5. tan x = sin x / cos x = (-4/5) / (-3/5) = 4/3.
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Which trigonometric functions are positive in the fourth quadrant?
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06 · Explore
Sum and Difference of Two Angles
Identities for the sum and difference of angles allow us to calculate values for non-standard angles.
The core identities are: cos(x + y) = cos x cos y - sin x sin y and cos(x - y) = cos x cos y + sin x sin y. Note the sign change in the cosine formulas.
For sine: sin(x + y) = sin x cos y + cos x sin y and sin(x - y) = sin x cos y - cos x sin y. For tangent: tan(x + y) = (tan x + tan y) / (1 - tan x tan y).
Calculating sin 15°
sin(45° - 30°)
Use sin(x - y) = sin x cos y - cos x sin y. sin 45° cos 30° - cos 45° sin 30° = (1/√2)(√3/2) - (1/√2)(1/2) = (√3 - 1) / (2√2).
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Expand cos(π/2 + x).
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07 · Explore
Multiple Angle Formulas
These formulas express trigonometric functions of 2x or 3x in terms of functions of x.
Double angle formulas: sin 2x = 2 sin x cos x. cos 2x has multiple forms: cos² x - sin² x, 2 cos² x - 1, or 1 - 2 sin² x. Also, tan 2x = 2 tan x / (1 - tan² x).
Triple angle formulas: sin 3x = 3 sin x - 4 sin³ x and cos 3x = 4 cos³ x - 3 cos x. These are useful for solving higher-degree trigonometric equations.
Proving sin 2x in terms of tan x
sin 2x = 2 tan x / (1 + tan² x)
Start with 2 sin x cos x. Divide by 1 (which is cos² x + sin² x). Divide numerator and denominator by cos² x to get the tangent form.
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Question
If sin x = 1/2, find sin 3x.
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08 · Explore
Sum-to-Product and Product-to-Sum Formulas
These identities transform the sum of two trigonometric functions into a product, or vice versa.
Sum-to-Product: sin x + sin y = 2 sin((x+y)/2) cos((x-y)/2). cos x + cos y = 2 cos((x+y)/2) cos((x-y)/2). cos x - cos y = -2 sin((x+y)/2) sin((x-y)/2). sin x - sin y = 2 cos((x+y)/2) sin((x-y)/2).
Product-to-Sum: 2 cos x cos y = cos(x + y) + cos(x - y). -2 sin x sin y = cos(x + y) - cos(x - y). These are essential for simplifying complex trigonometric expressions and integrals.
Simplifying a Ratio
(cos 7x + cos 5x) / (sin 7x - sin 5x)
Numerator = 2 cos 6x cos x using cos x + cos y. Denominator = 2 cos 6x sin x using sin x - sin y. Ratio = (2 cos 6x cos x) / (2 cos 6x sin x) = cos x / sin x = cot x.
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Express sin 5x + sin 3x as a product.
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