Mathematics › 🌌 Grade 11 › Trigonometric Identities & Solving Trigonometric Equations

Trigonometric Identities & Solving Trigonometric Equations

An identity holds for every angle; an equation holds only for the angles you are hunting — and because sine and cosine repeat, that hunt almost always turns up more than one answer.

Grade 11Algebra 2 / Pre-Calculus

Trigonometric Identities & Solving Trigonometric Equations — illustration

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The big idea: Every point on the unit circle satisfies x² + y² = 1, which is exactly sin²θ + cos²θ = 1 in disguise. From that one Pythagorean identity, the reciprocal, quotient, sum, difference, and double-angle identities all follow, and they let you rewrite a trig expression in whatever form is most useful. Solving a trig equation is a different task from proving an identity: an identity is a rewriting rule true for every θ, while an equation asks which particular θ make a statement true — and because sine and cosine are periodic, that search almost always turns up more than one angle.

🎯 By the end, you'll be able to

Derive and apply the Pythagorean, reciprocal, and quotient identities
Apply the angle-sum, difference, and double-angle identities
Solve a trigonometric equation for all solutions in [0, 2π) by finding both angles
Write the general solution of a trigonometric equation using + 2πn

📎 You should already know

Special Angles & the Unit Circle (advanced)
The Unit Circle

Where the Pythagorean identity comes from

On the unit circle, an angle θ lands at the point (cosθ, sinθ). Every point on the unit circle satisfies x² + y² = 1 by definition — that is what makes it the unit circle. Substitute the coordinates in and you get the single most useful identity in trigonometry.

Dividing that identity through by cos²θ or by sin²θ produces two more useful forms: 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. All three say the same underlying thing about the unit circle, just rearranged for whichever functions appear in the problem.

\[ \sin^{2}\theta + \cos^{2}\theta = 1 \]

The Pythagorean identity: true for every angle θ, because it is just x² + y² = 1 for a point on the unit circle.

Reciprocal and quotient identities

Secant, cosecant, and cotangent are defined as reciprocals: secθ = 1/cosθ, cscθ = 1/sinθ, cotθ = 1/tanθ. And tangent itself is a ratio of the other two: tanθ = sinθ/cosθ. These are definitions, not derived facts, but they are used constantly to rewrite an expression down to just sine and cosine — usually the easiest form to simplify.

\[ \tan\theta = \dfrac{\sin\theta}{\cos\theta}, \qquad \sec\theta=\dfrac{1}{\cos\theta},\ \ \csc\theta=\dfrac{1}{\sin\theta} \]

The quotient identity for tangent and the reciprocal identities for secant and cosecant.

Sum, difference, and double-angle identities

The angle-sum and difference identities expand sin or cos of a combined angle into sines and cosines of the pieces. Setting the two angles equal in the sum identities produces the double-angle identities as a special case — they are not new facts, just the sum identity used on θ + θ.

\[ \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta, \qquad \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta \]

Angle-sum and difference identities for sine and cosine (note the sign flip in the cosine identity).

\[ \sin(2\theta) = 2\sin\theta\cos\theta, \qquad \cos(2\theta) = \cos^{2}\theta - \sin^{2}\theta \]

Double-angle identities: the sum identities with α = β = θ.

✨ Identity versus equation — the line this lesson lives on

An identity, like sin²θ + cos²θ = 1, is true for every angle θ — it is a rewriting rule. An equation, like sinθ = ½, is only true for particular angles — it is a question to answer. Confusing the two leads to trying to “solve” an identity (there is nothing to solve; it is already always true) or trying to “simplify” an equation as if it held everywhere. One more notation trap along the way: sin²θ means (sinθ)² — square the output of sine — never the sine of θ².

Solving sin θ = k: there are usually two answers

A calculator's sin⁻¹ (arcsin) button returns exactly one angle. But on the unit circle, a horizontal line at height k typically crosses the circle at two points in one full turn, since sine is positive in both Quadrant I and Quadrant II (and negative in both III and IV). If θ₁ = arcsin(k) is the calculator's answer, the second solution is always θ₂ = π − θ₁.

🎮 Solve on the Circle LIVE

Drag the target line y = k across the unit circle and watch both intersection angles appear together.

⚠️ This is where students lose marks

Reporting only θ₁ = arcsin(k) and stopping is the single most common error in solving trig equations. Over [0, 2π), a typical sine or cosine equation has two solutions, not one. Always ask: where else on the circle does the function take this value?

🔑 The boundary cases

sinθ = 1 has exactly one solution in [0, 2π) — θ = π/2 — because the line y = 1 is tangent to the circle at a single point, not a true crossing. If |k| > 1, sinθ = k has no solution at all, since sine never leaves [−1, 1]. Beyond one full turn, a general solution adds whole revolutions: θ + 2πn for any integer n.

📝 Worked example: Solve sin θ = 1/2 for all θ in [0, 2π).

The reference angle is \( \theta_1 = \arcsin(1/2) = \pi/6 \).
Sine is positive in Quadrants I and II, so the second solution is \( \theta_2 = \pi - \pi/6 = 5\pi/6 \).
Both lie in [0, 2π), and no other angle in that range gives sin θ = 1/2.

✓ θ = π/6 and θ = 5π/6.

📝 Worked example: Solve 2cos²θ − 1 = 0 for all θ in [0, 2π).

Isolate: \( \cos^2\theta = \tfrac12 \), so \( \cos\theta = \pm\dfrac{1}{\sqrt2} \) — squaring means the square root reintroduces a ± sign.
For \( \cos\theta = \tfrac{1}{\sqrt2} \): reference angle π/4, and cosine is positive in QI and QIV, giving \( \theta = \pi/4 \) and \( \theta = 7\pi/4 \).
For \( \cos\theta = -\tfrac{1}{\sqrt2} \): cosine is negative in QII and QIII, giving \( \theta = 3\pi/4 \) and \( \theta = 5\pi/4 \).

✓ θ = π/4, 3π/4, 5π/4, 7π/4 — four solutions because the squared equation hides two separate cosine values.

Check your understanding

1. What is sin²θ + cos²θ for any angle θ?

The Pythagorean identity holds for every angle: it comes directly from x² + y² = 1 on the unit circle, so it is always 1.

2. Which expression correctly rewrites tan θ?

Tangent is defined as the quotient identity tan θ = sin θ / cos θ.

3. sin θ = 0.6 has how many solutions in [0, 2π)?

Since 0 < 0.6 < 1, the horizontal line y = 0.6 crosses the unit circle twice in one revolution — once in QI, once in QII.

4. sin θ = 1.5 has how many real solutions?

Sine never exceeds 1 in absolute value, so a target line at y = 1.5 never touches the unit circle — there is no solution.

5. sin²θ is best read as:

The exponent applies to the output of sine, not to θ itself: sin²θ = (sin θ)².

✅ Key takeaways

The Pythagorean identity sin²θ + cos²θ = 1 comes directly from x² + y² = 1 on the unit circle, and holds for every angle.
Reciprocal and quotient identities (sec, csc, cot, and tan θ = sin θ/cos θ) let you rewrite any trig expression in terms of sine and cosine.
Angle-sum, difference, and double-angle identities expand sin or cos of a combined angle; double-angle is just the sum identity with α = β.
An identity is true for every θ; an equation is true only for specific θ — solving means finding those specific angles.
sin θ = k typically has TWO solutions in [0, 2π) — θ₁ = arcsin k and θ₂ = π − θ₁ — with exactly one at the boundary k = ±1 and none when |k| > 1.