Trigonometry: Beyond Right Angles

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In Grade 10, trig lived inside the right-angled triangle. In Grade 11, we break free: angles can be $150°$, $240°$, even $-30°$ — and we can still calculate their sin, cos, and tan values. We also learn to solve any triangle (not just right-angled ones) and to prove identities.

The Four Big Ideas

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1. CAST Diagram & Reduction Formulas

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The CAST diagram tells you which ratios are positive in each quadrant:

Reduction means converting any angle back to an acute angle ($0°$ to $90°$) using the CAST diagram’s sign rules.

2. Trigonometric Identities

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The two fundamental identities you must know:

Every identity proof uses these two. The strategy: convert everything to $\sin$ and $\cos$, then simplify.

3. Trig Equations & General Solutions

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Solving $\sin\theta = 0.5$ doesn’t give just one answer — it gives infinitely many. The general solution captures all of them using $+ n \cdot 360°$ (or $+ n \cdot 180°$ for tan).

4. Sine Rule, Cosine Rule & Area Rule

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For any triangle (not just right-angled):

The Trig Graph Shapes

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In Grade 11, you also sketch trig functions with amplitude, period, and vertical shifts. Here are the basic shapes:

$y = \sin\theta$

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• Period: $360°$. Amplitude: 1. Range: $[-1;\, 1]$.
• Starts at the origin, peaks at $90°$, crosses zero at $180°$, troughs at $270°$.

$y = \cos\theta$

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• Period: $360°$. Amplitude: 1. Range: $[-1;\, 1]$.
• Starts at maximum (1), crosses zero at $90°$, troughs at $180°$.
• $\cos\theta$ is just $\sin\theta$ shifted left by $90°$.

$y = \tan\theta$

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• Period: $180°$ (NOT $360°$!). No amplitude (range is all real numbers).
• Asymptotes at $90°$, $270°$, etc. — the curve shoots to $\pm\infty$.
• Passes through the origin and through every multiple of $180°$.

Deep Dives (click into each)

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• Reduction Formulas & CAST — complete CAST explanation, all reduction formulas, co-functions, negative angles, worked examples
• Trig Identities & Equations — fundamental identities, proof strategies, general solutions, worked examples
• Sine, Cosine & Area Rules — decision flowchart, ambiguous case, 2D problem solving
• Trigonometric Graphs — sketching $y = a\sin(bx + p) + q$, amplitude, period, phase shift, reading information from graphs

🚨 Common Mistakes

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1. Calculator in wrong mode: Must be in DEG, not RAD. Check before every calculation.
2. Missing the second solution: $\sin\theta = 0.5$ gives $\theta = 30°$ AND $\theta = 150°$. Forgetting the second quadrant loses half the marks.
3. Reduction sign errors: Always check the CAST diagram for the sign. $\sin(180° + \theta) = -\sin\theta$, not $+\sin\theta$.
4. Dividing by $\sin\theta$ or $\cos\theta$: This loses solutions where they equal zero. Factor instead.
5. Tan period is $180°$, not $360°$: The general solution for tan uses $+ n \cdot 180°$.

🔗 Related Grade 11 topics:

• Functions — trig graphs use the same domain/range/transformation language
• Quadratic Equations — trig equations often become quadratics
• Analytical Geometry: Inclination — $\tan\theta = m$ connects trig to gradients

📌 Grade 10 foundation: Trig Ratios & Special Angles

📌 Grade 12 extension: Trigonometry — compound angles, double angles, and more complex identities

⏮️ Probability | 🏠 Back to Grade 11 | ⏭️ Circle Geometry