Trigonometry & Calculus

Trigonometric Laws and Identities

Grade 10 · Grade 11 · Grade 12

✓Apply the Sine Law to solve for unknown sides and angles in non-right triangles.
✓Apply the Cosine Law to solve for unknown sides and angles in non-right triangles.
✓State and apply the fundamental trigonometric identities (reciprocal, quotient, Pythagorean).
✓Simplify trigonometric expressions and prove trigonometric identities.

Key concepts

Sine Law

The Sine Law describes the relationship between the sides and angles of any triangle (not just right triangles). It states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides of the triangle. It is used when you know an angle and its opposite side, along with one other angle or side.

a/sin A = b/sin B = c/sin C

Cosine Law

The Cosine Law is a generalization of the Pythagorean theorem that relates the lengths of the sides of any triangle to the cosine of one of its angles. It is used when you know two sides and the contained angle (SAS) or all three sides (SSS) of a triangle.

a^2 = b^2 + c^2 - 2bc cos A

Reciprocal Identities

These identities define the reciprocal trigonometric ratios (cosecant, secant, cotangent) in terms of the primary ratios (sine, cosine, tangent).

csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ

Quotient Identities

These identities express tangent and cotangent in terms of sine and cosine, which is often useful for simplifying expressions or proving other identities.

tan θ = sin θ / cos θ, cot θ = cos θ / sin θ

Pythagorean Identities

These fundamental identities are derived from the Pythagorean theorem and the unit circle. They are crucial for simplifying trigonometric expressions and proving identities.

sin^2 θ + cos^2 θ = 1, 1 + tan^2 θ = sec^2 θ, 1 + cot^2 θ = csc^2 θ

Key facts to remember

1Sine Law: a/sin A = b/sin B = c/sin C
2Cosine Law: a^2 = b^2 + c^2 - 2bc cos A (and its permutations for b^2 and c^2)
3Pythagorean Identity: sin^2 θ + cos^2 θ = 1
4Quotient Identity: tan θ = sin θ / cos θ
5Reciprocal Identities: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
6The ambiguous case of the Sine Law occurs when given SSA (Side-Side-Angle), which may result in zero, one, or two possible triangles.
7When proving identities, it's often helpful to convert all terms to sine and cosine.

Worked examples

Example 1

In triangle ABC, angle A = 45°, angle B = 60°, and side a = 10 cm. Find the length of side b to one decimal place.

IState the Sine Law: a/sin A = b/sin B

IISubstitute the known values: 10/sin 45° = b/sin 60°

IIIIsolate b: b = (10 * sin 60°) / sin 45°

IVCalculate the value: b = (10 * 0.8660) / 0.7071

Vb ≈ 12.247

Answer

Side b ≈ 12.2 cm

Ensure your calculator is in degree mode.

Example 2

In triangle XYZ, side x = 7 m, side y = 9 m, and side z = 12 m. Find angle Z to the nearest degree.

IState the Cosine Law for angle Z: z^2 = x^2 + y^2 - 2xy cos Z

IISubstitute the known values: 12^2 = 7^2 + 9^2 - 2(7)(9) cos Z

IIISimplify: 144 = 49 + 81 - 126 cos Z

IVRearrange to solve for cos Z: 144 = 130 - 126 cos Z

V14 = -126 cos Z

VIcos Z = 14 / -126

VIIcos Z ≈ -0.1111

VIIICalculate Z using the inverse cosine function: Z = cos⁻¹(-0.1111)

9Z ≈ 96.38°

Answer

Angle Z ≈ 96°

When finding an angle using the Cosine Law, it's possible to get an obtuse angle (greater than 90°), which will have a negative cosine value.

Example 3

Prove the identity: (sin θ / (1 + cos θ)) + ((1 + cos θ) / sin θ) = 2 csc θ

IStart with the Left Side (LS): LS = sin θ / (1 + cos θ) + (1 + cos θ) / sin θ

IIFind a common denominator, which is sin θ(1 + cos θ):

IIILS = (sin θ * sin θ) / (sin θ(1 + cos θ)) + ((1 + cos θ)(1 + cos θ)) / (sin θ(1 + cos θ))

IVLS = (sin^2 θ + (1 + cos θ)^2) / (sin θ(1 + cos θ))

VExpand the numerator: LS = (sin^2 θ + 1 + 2cos θ + cos^2 θ) / (sin θ(1 + cos θ))

VIGroup sin^2 θ and cos^2 θ: LS = ((sin^2 θ + cos^2 θ) + 1 + 2cos θ) / (sin θ(1 + cos θ))

VIIApply the Pythagorean Identity (sin^2 θ + cos^2 θ = 1): LS = (1 + 1 + 2cos θ) / (sin θ(1 + cos θ))

VIIISimplify the numerator: LS = (2 + 2cos θ) / (sin θ(1 + cos θ))

9Factor out 2 from the numerator: LS = 2(1 + cos θ) / (sin θ(1 + cos θ))

10Cancel the common factor (1 + cos θ): LS = 2 / sin θ

11Apply the Reciprocal Identity (1/sin θ = csc θ): LS = 2 csc θ

12This matches the Right Side (RS).

Answer

LS = 2 csc θ = RS. The identity is proven.

When proving identities, always work on one side (usually the more complex side) until it transforms into the other side. Do not work on both sides simultaneously.

Common mistakes

✗Using the Sine Law when the Cosine Law is required (e.g., for SAS or SSS triangles).
✗Incorrectly applying the ambiguous case of the Sine Law, leading to missing a possible solution or including an impossible one.
✗Algebraic errors when rearranging the Cosine Law to solve for an angle.
✗Forgetting to square the trigonometric function (e.g., writing sin θ^2 instead of sin^2 θ).
✗Attempting to work on both sides of an identity simultaneously without showing a clear logical progression from one side to the other.
✗Not checking calculator mode (degrees vs. radians) for triangle problems.

Exam tips

★Always draw a diagram for triangle problems to visualize the given information and what needs to be found.
★Memorize the fundamental trigonometric identities; they are building blocks for more complex problems.
★When proving identities, start with the more complex side and simplify it step-by-step until it matches the other side.
★If you get stuck proving an identity, try converting all expressions to sine and cosine.

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