Pre-Calculus

Advanced Trigonometry: Identities, Laws, and Polar Coordinates

Pre-Calculus

✓Simplify trigonometric expressions and prove trigonometric identities using fundamental identities.
✓Solve oblique triangles using the Law of Sines and the Law of Cosines.
✓Convert coordinates and equations between rectangular and polar forms.
✓Graph points and basic equations in the polar coordinate system.

Key concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables for which the functions are defined. They are crucial tools for simplifying complex trigonometric expressions, solving trigonometric equations, and proving other mathematical relationships. Mastering these identities is fundamental for advanced work in calculus and physics.

* **Reciprocal Identities:**\n * sin θ = 1/csc θ\n * cos θ = 1/sec θ\n * tan θ = 1/cot θ\n * csc θ = 1/sin θ\n * sec θ = 1/cos θ\n * cot θ = 1/tan θ\n * **Quotient Identities:**\n * tan θ = sin θ / cos θ\n * cot θ = cos θ / sin θ\n * **Pythagorean Identities:**\n * sin² θ + cos² θ = 1\n * 1 + tan² θ = sec² θ\n * 1 + cot² θ = csc² θ\n * **Even/Odd Identities:**\n * sin(-θ) = -sin θ\n * cos(-θ) = cos θ\n * tan(-θ) = -tan θ\n * **Cofunction Identities:**\n * sin(π/2 - θ) = cos θ\n * cos(π/2 - θ) = sin θ\n * tan(π/2 - θ) = cot θ\n * *Note: Sum/Difference, Double/Half-Angle identities are also important but are often introduced after these fundamental ones. For this lesson, we focus on the foundational set.*

Law of Sines

The Law of Sines is used to solve oblique triangles (triangles that do not contain a right angle). It establishes a relationship between the sides of a triangle and the sines of their opposite angles. It is particularly useful when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA, which is the ambiguous case).

For a triangle with angles A, B, C and opposite sides a, b, c respectively:\n a / sin A = b / sin B = c / sin C

Law of Cosines

The Law of Cosines is another tool for solving oblique triangles. It is an extension of the Pythagorean theorem and is used when you know two sides and their included angle (SAS), or all three sides (SSS). It helps find missing side lengths or angle measures when the Law of Sines cannot be directly applied.

For a triangle with angles A, B, C and opposite sides a, b, c respectively:\n * a² = b² + c² - 2bc cos A\n * b² = a² + c² - 2ac cos B\n * c² = a² + b² - 2ab cos C

Polar Coordinates

Polar coordinates offer an alternative way to locate points in a plane, using a distance from the origin (pole) and an angle from the positive x-axis (polar axis). A point is represented by (r, θ), where 'r' is the directed distance from the pole to the point, and 'θ' is the directed angle from the polar axis to the line segment connecting the pole to the point. This system is especially useful for describing circular motion or objects with radial symmetry.

* **Conversion from Polar (r, θ) to Rectangular (x, y):**\n * x = r cos θ\n * y = r sin θ\n * **Conversion from Rectangular (x, y) to Polar (r, θ):**\n * r² = x² + y² (or r = ±√(x² + y²))\n * tan θ = y / x (when x ≠ 0)\n * *Note: When converting from rectangular to polar, careful consideration of the quadrant of (x, y) is needed to determine the correct angle θ.*

Key facts to remember

1Fundamental trigonometric identities (Reciprocal, Quotient, Pythagorean) are essential for simplifying expressions and proving other identities.
2The Law of Sines (a/sin A = b/sin B = c/sin C) is used for AAS, ASA, and SSA (ambiguous case) triangles.
3The Law of Cosines (e.g., a² = b² + c² - 2bc cos A) is used for SAS and SSS triangles.
4To convert from polar (r, θ) to rectangular (x, y): x = r cos θ, y = r sin θ.
5To convert from rectangular (x, y) to polar (r, θ): r² = x² + y², tan θ = y/x. Be mindful of the quadrant for θ.
6Polar coordinates (r, θ) can have multiple representations for the same point, e.g., (r, θ) and (-r, θ + π).

Worked examples

Example 1

Prove the identity: (sec x - cos x) / sin x = tan x

IStart with the left-hand side (LHS) of the identity.

IILHS = (sec x - cos x) / sin x

IIIRewrite sec x in terms of cos x: sec x = 1/cos x.

IVLHS = (1/cos x - cos x) / sin x

VFind a common denominator for the numerator: 1/cos x - cos x = (1 - cos² x) / cos x.

VILHS = [(1 - cos² x) / cos x] / sin x

VIIUse the Pythagorean identity sin² x + cos² x = 1, so 1 - cos² x = sin² x.

VIIILHS = (sin² x / cos x) / sin x

9Simplify the complex fraction: (sin² x / cos x) * (1 / sin x)

10Cancel out one sin x term: sin x / cos x

11Rewrite in terms of tan x: sin x / cos x = tan x.

12LHS = tan x = RHS. The identity is proven.

Answer

The identity is proven by transforming the left-hand side into the right-hand side using fundamental trigonometric identities.

Always start with one side (usually the more complex one) and work towards the other side. Avoid working on both sides simultaneously.

Example 2

In triangle ABC, a = 10, b = 15, and angle C = 110°. Find the length of side c and the measures of angles A and B. Round side lengths to one decimal place and angles to the nearest degree.

IWe are given two sides (a, b) and the included angle (C), which is an SAS case. We must use the Law of Cosines to find side c first.

IIc² = a² + b² - 2ab cos C

IIIc² = 10² + 15² - 2(10)(15) cos 110°

IVc² = 100 + 225 - 300(-0.3420...)

Vc² = 325 + 102.60

VIc² = 427.60

VIIc = √427.60 ≈ 20.68 ≈ 20.7

VIIINow that we have side c, we can use the Law of Sines to find angle A or B. Let's find angle A.

9a / sin A = c / sin C

1010 / sin A = 20.68 / sin 110°

11sin A = (10 * sin 110°) / 20.68

12sin A = (10 * 0.9397) / 20.68

13sin A = 9.397 / 20.68 ≈ 0.4544

14A = arcsin(0.4544) ≈ 27.03° ≈ 27°

15Finally, find angle B using the fact that the sum of angles in a triangle is 180°.

16B = 180° - A - C

17B = 180° - 27° - 110°

18B = 43°

Answer

Side c ≈ 20.7, Angle A ≈ 27°, Angle B ≈ 43°.

When using the Law of Sines after the Law of Cosines, it's generally safer to find the smallest remaining angle first (the one opposite the smallest side) to avoid the ambiguous case if you were to use the Law of Sines to find an obtuse angle.

Example 3

a) Convert the rectangular point (-3, 3√3) to polar coordinates (r, θ) with r > 0 and 0 ≤ θ < 2π. b) Convert the polar equation r = 4 sin θ to its rectangular form.

Ia) For the point (-3, 3√3):

IICalculate r: r² = x² + y² = (-3)² + (3√3)² = 9 + (9 * 3) = 9 + 27 = 36.

IIISo, r = √36 = 6 (since r > 0).

IVCalculate θ: tan θ = y / x = (3√3) / (-3) = -√3.

VSince x is negative and y is positive, the point is in Quadrant II. The reference angle for tan θ = √3 is π/3 (or 60°).

VIIn Quadrant II, θ = π - π/3 = 2π/3 (or 180° - 60° = 120°).

VIISo, the polar coordinates are (6, 2π/3).

VIIIb) For the polar equation r = 4 sin θ:

9Recall the conversion formulas: x = r cos θ, y = r sin θ, r² = x² + y².

10Multiply both sides of the equation by r to introduce r² and r sin θ terms:

11r * r = 4 sin θ * r

12r² = 4r sin θ

13Substitute r² = x² + y² and r sin θ = y:

14x² + y² = 4y

15Rearrange to standard form for a circle (optional, but good practice):

16x² + y² - 4y = 0

17Complete the square for y: x² + (y² - 4y + 4) = 4

18x² + (y - 2)² = 4

Answer

a) The polar coordinates are (6, 2π/3). b) The rectangular equation is x² + (y - 2)² = 4 (a circle centered at (0, 2) with radius 2).

When converting from rectangular to polar, always check the quadrant of the original rectangular point to ensure the correct angle θ is chosen, as tan θ has a period of π.

Common mistakes

✗Incorrectly applying or misremembering trigonometric identities, especially signs in Pythagorean identities.
✗Failing to recognize or correctly handle the ambiguous case (SSA) when using the Law of Sines, which can lead to two possible triangles, one triangle, or no triangle.
✗Not considering the quadrant of a rectangular point when converting to polar coordinates, leading to an incorrect angle θ.
✗Confusing the roles of r and θ in polar coordinates or x and y in rectangular coordinates during conversions.
✗Assuming that a triangle is a right triangle when it is not, and attempting to use SOH CAH TOA instead of the Law of Sines or Cosines.

Exam tips

★Memorize the fundamental trigonometric identities. Practice deriving less common ones from the fundamental set.
★Always draw a diagram for triangle problems to visualize the given information and the unknown quantities. This helps in choosing the correct law (Sines or Cosines).
★When converting between coordinate systems, clearly write down the conversion formulas before substituting values.
★For polar coordinates, be comfortable with angles in both degrees and radians, and understand how negative r values affect point location.

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