Express $\sin 54^\circ + \cos 72^\circ $ in terms of Trigonometric ratios of angles between $ 0^\circ $ and $ 45^\circ $.

The correct answer is Option D: $\cos 36^\circ + \sin 18^\circ$. Use the co-function identity: convert sin to cos by subtracting the angle from 90°. Thus, sin 54° becomes cos(90°-54°) = cos 36°, and cos 72° becomes sin(90°-72°) = sin 18°. Step-by-Step Solution & Concepts: Shortcut Trick: Use the co-function identity: convert sin to cos by subtracting the angle from 90°. Thus, sin 54° becomes cos(90°-54°) = cos 36°, and cos 72° becomes sin(90°-72°) = sin 18°. Formula Used: $$\sin(\theta) = \cos(90^\circ - \theta)$$ Given: Expression sin 54° + cos 72°. The objective is to express this in terms of trigonometric ratios of angles between 0° and 45°. Steps: Apply Co-function Identity to sin 54°: Using the identity $\sin(\theta) = \cos(90^\circ - \theta)$, substitute $\theta = 54^\circ$.. Apply Co-function Identity to cos 72°: Using the identity $\cos(\theta) = \sin(90^\circ - \theta)$, substitute $\theta = 72^\circ$.. Combine results: Adding the two converted terms, we get $\cos(36^\circ) + \sin(18^\circ)$. This matches the condition that the angles (36° and 18°) are between 0° and 45°. Answer: D is correct because applying co-function identities transforms $\sin 54^\circ$ to $\cos 36^\circ$ and $\cos 72^\circ$ to $\sin 18^\circ$, resulting in $\cos 36^\circ + \sin 18^\circ$. Exam Tip: This concept of complementary angles is foundational for evaluating trigonometric series and simplifying complex expressions in advanced calculus.

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MathematicsPrevious Year Paper 2024

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Express $\sin 54°+ \cos 72°$ in terms of Trigonometric ratios of angles between $0°$ and $45°$ .

$\cos 36°+ \cos 18°$

$\sin 36°+ \sin 18°$

$\sin 36°+ \cos 18°$

$\cos 36°+ \sin 18°$

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option D

$\cos 36°+ \sin 18°$

Quick Summary:

Use the co-function identity: convert sin to cos by subtracting the angle from 90°. Thus, sin 54° becomes cos(90°-54°) = cos 36°, and cos 72° becomes sin(90°-72°) = sin 18°.

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📐MAMath SolutionDirect Formula

📋 Given

Expression sin 54° + cos 72°. The objective is to express this in terms of trigonometric ratios of angles between 0° and 45°.

🔢 Formula Used

sin⁡(θ)=cos⁡(90°−θ)\sin(\theta) = \cos(90°- \theta)sin(θ)=cos(90°−θ)

⚡ Exam Hall Shortcut / Speed Trick

Use the co-function identity: convert sin to cos by subtracting the angle from 90°. Thus, sin 54° becomes cos(90°-54°) = cos 36°, and cos 72° becomes sin(90°-72°) = sin 18°.

⚠️ Common Student Trap / Pitfall

Confusing the range requirement (0°-45°) or incorrectly applying the co-function identity by subtracting from 180° instead of 90°.

🔢 Step-by-Step Solution

Apply Co-function Identity to sin 54°

Using the identity $\sin(\theta) = \cos(90°- \theta)$ , substitute $\theta = 54°$ .

sin⁡(54°)=cos⁡(90°−54°)=cos⁡(36°)\sin(54°) = \cos(90°- 54°) = \cos(36°)sin(54°)=cos(90°−54°)=cos(36°)

Apply Co-function Identity to cos 72°

Using the identity $\cos(\theta) = \sin(90°- \theta)$ , substitute $\theta = 72°$ .

cos⁡(72°)=sin⁡(90°−72°)=sin⁡(18°)\cos(72°) = \sin(90°- 72°) = \sin(18°)cos(72°)=sin(90°−72°)=sin(18°)

Combine results

Adding the two converted terms, we get $\cos(36°) + \sin(18°)$ . This matches the condition that the angles (36° and 18°) are between 0° and 45°.

cos⁡(36°)+sin⁡(18°)\cos(36°) + \sin(18°)cos(36°)+sin(18°)

✅

D is correct because applying co-function identities transforms $\sin 54°$ to $\cos 36°$ and $\cos 72°$ to $\sin 18°$ , resulting in $\cos 36°+ \sin 18°$ .

Core Concepts Used

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Trigonometric Identities Complementary Angles Co-function Formulas

💡 EXAM TIP

This concept of complementary angles is foundational for evaluating trigonometric series and simplifying complex expressions in advanced calculus.

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