Sec 1 - Mathematics
Unit 2: Trigonometry and Plane Geometry
Analytical Trigonometry: Unit Circle Equation, Primary Functions, and Reciprocal Ratios
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The unit circle (x² + y² = 1) centered at the origin is the ultimate mathematical instrument defining trigonometric functions for any real angle θ. When the terminal ray of θ intersects the unit circle at point P(x, y), x represents cos θ, y represents sin θ, and y/x defines tan θ (for x ≠ 0). From these fundamental definitions emerge the reciprocal ratios: sec θ = 1/x, csc θ = 1/y, and cot θ = x/y. We derive the Pythagorean identity sin² θ + cos² θ = 1 and analyze sign distribution across all four quadrants (All-Students-Take-Calculus rule).
1. Unit Circle & Coordinates
Fundamental definitions:
- Unit Circle: x² + y² = 1.
- Intersection Point P(x, y):
• cos θ = x
• sin θ = y
• tan θ = y / x = sin θ / cos θ (x ≠ 0) - Pythagorean Identity: sin² θ + cos² θ = 1.
2. Reciprocals & Quadrant Signs
Reciprocals and signs rule:
- Reciprocal Ratios:
• sec θ = 1 / cos θ = 1 / x
• csc θ = 1 / sin θ = 1 / y
• cot θ = 1 / tan θ = x / y - Quadrant Signs:
• Quadrant I: All functions positive (+).
• Quadrant II: sin and csc positive (+).
• Quadrant III: tan and cot positive (+).
• Quadrant IV: cos and sec positive (+).
Worked Application:
Problem: Point P(-4/5, y) lies on the unit circle in Quadrant II. Find all trigonometric ratios for θ.
Solution: x² + y² = 1 ⇒ (-4/5)² + y² = 1 ⇒ 16/25 + y² = 1 ⇒ y² = 9/25 ⇒ y = 3/5 (since y > 0 in Quad II).
cos θ = -4/5, sin θ = 3/5, tan θ = (3/5) / (-4/5) = -3/4.
Solution: x² + y² = 1 ⇒ (-4/5)² + y² = 1 ⇒ 16/25 + y² = 1 ⇒ y² = 9/25 ⇒ y = 3/5 (since y > 0 in Quad II).
cos θ = -4/5, sin θ = 3/5, tan θ = (3/5) / (-4/5) = -3/4.
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unit circle
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دائرة الوحدة
📖 Def: circle with center at origin (0, 0) and radius r = 1
sine
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جيب الزاوية (جا)
📖 Def: the y-coordinate of terminal point P on the unit circle
cosine
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جيب تمام الزاوية (جتا)
📖 Def: the x-coordinate of terminal point P on the unit circle
tangent
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ظل الزاوية (ظا)
📖 Def: ratio of sine to cosine: y / x
cosecant
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قاطع التمام (قتا)
📖 Def: reciprocal of sine: 1 / sin θ
secant
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قاطع الزاوية (قا)
📖 Def: reciprocal of cosine: 1 / cos θ
cotangent
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ظل التمام (ظتا)
📖 Def: reciprocal of tangent: 1 / tan θ
Pythagorean identity
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متطابقة فيثاغورس المثلثية
📖 Def: sin² θ + cos² θ = 1
Click on an option to check your answer and view the detailed explanation:
1
If terminal point P(x, 12/13) is on the unit circle with x < 0, the value of x is:
2
If sin θ < 0 and cos θ > 0, angle θ lies in quadrant:
3
The exact value of sin² 53° + cos² 53° is:
4
If csc θ = -2 and θ ∈ [π, 3π/2], then sin θ equals:
5
Point P on the unit circle corresponding to angle π (180°) has coordinates:
6
Simplifying the expression tan θ · cot θ + cos θ · sec θ yields:
7
If tan θ = 1 and θ ∈ [0, 2π], the two possible angle values are:
8
The range of function f(θ) = cos θ for all real numbers θ is:
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