Question

In a triangle \(\mathrm{ABC}\), if \(\mathrm{A}=45^{\circ}, \mathrm{C}=75^{\circ}\), then the side a is (a) \(\sqrt{2} \mathrm{~b}\) (b) \(\frac{2}{\sqrt{3}} \mathrm{~b}\) (c) \(\frac{\sqrt{2}}{\sqrt{3}} \mathrm{~b}\) (d) \(\frac{3}{\sqrt{2}} \mathrm{~b}\)

Short answer

In a triangle ABC, angle A = 45° and angle C = 75°. Find the correct formula for the ratio of side a to side b: a) \(\frac{a}{b} = \frac{\sqrt{3}}{\sqrt{2}}\) b) \(\frac{a}{b} = \frac{1}{\sqrt{3}}\) c) \(\frac{a}{b} = \frac{\sqrt{2}}{\sqrt{3}}\) d) \(\frac{a}{b} = \frac{1}{2}\) Solution: 1. Calculate angle B: B = 180° - 45° - 75° = 60° 2. Apply the sine rule: \(\frac{a}{b} = \frac{\sin 45^{\circ}}{\sin 60^{\circ}}\) 3. Simplify the equation with known values of sine: \(\frac{a}{b} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{3}}{2}} = \frac{\sqrt{2}}{\sqrt{3}}\) The correct formula for the ratio of side a to side b is: \(\frac{a}{b} = \frac{\sqrt{2}}{\sqrt{3}}\) (option c).

Step-by-step solution

Calculate Angle B

Since the sum of angles in a triangle is always 180 degrees, we can find angle B using the formula: B = 180° - A - C Plugging in the given values, we have: B = 180° - 45° - 75° = 60°

Apply the Sine Rule

The sine rule states that for any triangle with sides a, b, and c, and their opposite angles A, B, and C respectively: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\) We will use the sine rule to find the ratio \(\frac{a}{b}\). From the sine rule, we have: \(\frac{a}{\sin A} = \frac{b}{\sin B}\) Now, we'll solve for \(\frac{a}{b}\): \(\frac{a}{b} = \frac{\sin A}{\sin B}\)

Compare the Result with the Given Options

Plugging in the given values of A and B into the formula, we have: \(\frac{a}{b} = \frac{\sin 45^{\circ}}{\sin 60^{\circ}}\) We know that \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\) and \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\). Plugging these values into the formula, we get: \(\frac{a}{b} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{3}}{2}}\) Simplifying the equation, we get: \(\frac{a}{b} = \frac{\sqrt{2}}{\sqrt{3}}\) Comparing this result with the given options, we can see that the correct answer is option (c): \(\frac{a}{b} = \frac{\sqrt{2}}{\sqrt{3}}\)

Key concepts

Sine Rule

The Sine Rule is a valuable tool in trigonometry, especially when dealing with non-right angled triangles. It allows us to find unknown angles or sides with ease. The rule states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all sides of the triangle. Therefore, for a triangle with sides \(a\), \(b\), \(c\) opposite angles \(A\), \(B\), \(C\) respectively:

• \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)

To use the Sine Rule effectively, identify the known angles and sides of the triangle, and apply the formula to solve for the unknown quantities. This rule is very flexible as it is applicable in any type of triangle, be it acute, obtuse, or scalene.

Angle Sum Property of Triangle

The Angle Sum Property of a Triangle is a fundamental concept in geometry. It states that the sum of the interior angles of a triangle is always 180 degrees. This property is applicable to all triangles, regardless of their shape or size. To find a missing angle when two angles are known:

• Add the measures of the two known angles.
• Subtract the sum from 180 degrees to get the third angle.

For instance, if you have angles \(A = 45^\circ\) and \(C = 75^\circ\), you can find angle \(B\) by calculating \(B = 180^\circ - 45^\circ - 75^\circ = 60^\circ\). Recognizing and applying this property can simplify the process of working with triangles and is a crucial step in solving many geometric problems.

Trigonometric Ratios

Trigonometric Ratios are the ratios of the sides of a right-angled triangle and they are fundamental in the study of trigonometry. They are used extensively to solve geometric problems involving triangles. The primary trigonometric ratios are sine, cosine, and tangent. These ratios are defined as follows for a right triangle:

• Sine of angle \(\theta\) (\(\sin \theta\)) is the ratio of the side opposite to \(\theta\) to the hypotenuse: \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\).
• Cosine of angle \(\theta\) (\(\cos \theta\)) is the ratio of the adjacent side to \(\theta\) to the hypotenuse: \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\).
• Tangent of angle \(\theta\) (\(\tan \theta\)) is the ratio of the opposite side to the adjacent side: \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\).

These ratios are pivotal for finding angles and sides in right-angled triangles and are utilized in conjunction with inverse trigonometric functions for comprehensive problem-solving. Understanding these fundamental ratios provides the foundation for advanced studies in trigonometry.