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Trigonometry - Trigonometric Ratios of Complementary Angles

Grade 9ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Definition of Complementary Angles: Two angles are said to be complementary if their sum is exactly 90∘90^{\circ}. In a right-angled triangle, if one acute angle is θ\theta, the other acute angle must be 90∘−θ90^{\circ} - \theta because the sum of all angles in a triangle is 180∘180^{\circ}.

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Visualizing the Right-Angled Triangle: Imagine a triangle ABCABC with a right angle at BB. If you look from angle AA (let ∠A=θ\angle A = \theta), the side BCBC is the 'opposite' side. However, if you look from the complementary angle CC (where ∠C=90∘−θ\angle C = 90^{\circ} - \theta), that same side BCBC becomes the 'adjacent' side. This change in perspective is the basis for complementary ratios.

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Sine and Cosine Relationship: The sine of an angle is equal to the cosine of its complement. This means that the ratio of the opposite side to the hypotenuse for θ\theta is the same as the ratio of the adjacent side to the hypotenuse for 90∘−θ90^{\circ} - \theta. Thus, sin⁡θ=cos⁡(90∘−θ)\sin \theta = \cos(90^{\circ} - \theta).

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Tangent and Cotangent Relationship: The tangent of an angle is equal to the cotangent of its complement. In a triangle, the ratio of opposite/adjacent for one acute angle is the inverse (adjacent/opposite) for the other. Therefore, tan⁡θ=cot⁡(90∘−θ)\tan \theta = \cot(90^{\circ} - \theta).

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Secant and Cosecant Relationship: The secant of an angle is equal to the cosecant of its complement. This follows the same logic as the sine/cosine relationship, using the reciprocal ratios of the hypotenuse and the sides. So, sec⁡θ=csc⁡(90∘−θ)\sec \theta = \csc(90^{\circ} - \theta).

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The 'Co-' Prefix Meaning: The 'co' in Cosine, Cotangent, and Cosecant stands for 'complementary'. These ratios were named specifically to identify them as the sine, tangent, and secant of the complementary angle.

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Application in Simplification: Complementary angle formulas are primarily used to simplify expressions where the sum of the angles in the numerator and denominator (or adjacent terms) is 90∘90^{\circ}. For example, sin⁡20∘\sin 20^{\circ} can be replaced by cos⁡70∘\cos 70^{\circ} to cancel out terms in a fraction.

📐Formulae

sin⁡(90∘−θ)=cos⁡θ\sin (90^{\circ} - \theta) = \cos \theta

cos⁡(90∘−θ)=sin⁡θ\cos (90^{\circ} - \theta) = \sin \theta

tan⁡(90∘−θ)=cot⁡θ\tan (90^{\circ} - \theta) = \cot \theta

cot⁡(90∘−θ)=tan⁡θ\cot (90^{\circ} - \theta) = \tan \theta

sec⁡(90∘−θ)=csc⁡θ\sec (90^{\circ} - \theta) = \csc \theta

csc⁡(90∘−θ)=sec⁡θ\csc (90^{\circ} - \theta) = \sec \theta

💡Examples

Problem 1:

Evaluate: sin⁡36∘cos⁡54∘−tan⁡70∘cot⁡20∘\frac{\sin 36^{\circ}}{\cos 54^{\circ}} - \frac{\tan 70^{\circ}}{\cot 20^{\circ}}

Solution:

Step 1: Observe the angles. In the first term, 36∘+54∘=90∘36^{\circ} + 54^{\circ} = 90^{\circ}. In the second term, 70∘+20∘=90∘70^{\circ} + 20^{\circ} = 90^{\circ}. Step 2: Apply the complementary ratio formula to the numerators: sin⁡36∘=sin⁡(90∘−54∘)=cos⁡54∘\sin 36^{\circ} = \sin(90^{\circ} - 54^{\circ}) = \cos 54^{\circ} tan⁡70∘=tan⁡(90∘−20∘)=cot⁡20∘\tan 70^{\circ} = \tan(90^{\circ} - 20^{\circ}) = \cot 20^{\circ} Step 3: Substitute these back into the original expression: cos⁡54∘cos⁡54∘−cot⁡20∘cot⁡20∘\frac{\cos 54^{\circ}}{\cos 54^{\circ}} - \frac{\cot 20^{\circ}}{\cot 20^{\circ}} Step 4: Simplify the fractions: 1−1=01 - 1 = 0

Explanation:

This solution uses the complementary angle identities to convert the numerator of each fraction to match its denominator, allowing them to be simplified to 1.

Problem 2:

Find the value of θ\theta if sin⁡3θ=cos⁡(θ−6∘)\sin 3\theta = \cos(\theta - 6^{\circ}), where 3θ3\theta and (θ−6∘)(\theta - 6^{\circ}) are acute angles.

Solution:

Step 1: Use the identity cos⁡A=sin⁡(90∘−A)\cos A = \sin(90^{\circ} - A) to make the trigonometric ratios on both sides the same. sin⁡3θ=sin⁡(90∘−(θ−6∘))\sin 3\theta = \sin(90^{\circ} - (\theta - 6^{\circ})) Step 2: Since the sine ratios are equal and the angles are acute, we can equate the angles: 3θ=90∘−(θ−6∘)3\theta = 90^{\circ} - (\theta - 6^{\circ}) Step 3: Solve the linear equation: 3θ=90∘−θ+6∘3\theta = 90^{\circ} - \theta + 6^{\circ} 3θ+θ=96∘3\theta + \theta = 96^{\circ} 4θ=96∘4\theta = 96^{\circ} Step 4: Divide by 4: θ=96∘4=24∘\theta = \frac{96^{\circ}}{4} = 24^{\circ}

Explanation:

By converting the cosine term into a sine term using complementary angle properties, we can create an algebraic equation to solve for the unknown angle.