Question

\(\frac{\sin ^{2} 5^{\circ}+\sin ^{2} 10^{\circ}+\ldots+\sin ^{2} 85^{\circ}+\sin ^{2} 90^{\circ}}{1+\cos ^{2} 5^{\circ}+\cos ^{2} 10^{\circ}+\ldots+\cos ^{2} 85^{\circ}+\cos ^{2} 90^{\circ}}\) equals (a) 1 (b) \(\frac{19}{17}\) (c) 2 (d) \(\frac{17}{19}\)

Short answer

a) 1 b) $\frac{19}{17}$ c) 19 d) $\frac{17}{19}$ Answer: d) $\frac{17}{19}$

Step-by-step solution

Recognize the Angle Pairs

Notice that some angle pairs have a special relationship in this problem. If we consider the sine squared values at 5 degrees and 85 degrees, we can use the identity \(\sin^2(x)=\cos^2(90-x)\): \(\sin^2(5^{\circ}) = \cos^2(90-5^{\circ})=\cos^2(85^{\circ})\) Similar relationship can be found for other angle pairs like 10 and 80, 15 and 75, ... , 40 and 50 degrees.

Rewrite the expression

Based on the relationship found in step 1, we can rewrite the given expression: \(\frac{\sin^2(5^{\circ})+\sin^2(10^{\circ})+...+\sin^2(85^{\circ})+\sin^2(90^{\circ})}{1+\cos^2(5^{\circ})+\cos^2(10^{\circ})+...+\cos^2(85^{\circ})+\cos^2(90^{\circ})}=\frac{\cos^2(85^{\circ})+\cos^2(80^{\circ})+...+\cos^2(40^{\circ})+\sin^2(90^{\circ})}{1+\cos^2(5^{\circ})+\cos^2(10^{\circ})+...+\cos^2(85^{\circ})+\cos^2(90^{\circ})}\)

Combine terms

Now, we can combine terms in the numerator and the denominator: \(\frac{\cos^2(85^{\circ})+\cos^2(80^{\circ})+...+\cos^2(40^{\circ})+\sin^2(90^{\circ})}{1+\cos^2(5^{\circ})+\cos^2(10^{\circ})+...+\cos^2(85^{\circ})+\cos^2(90^{\circ})}=\frac{\cos^2(40^{\circ})+\cos^2(45^{\circ})+\cos^2(50^{\circ})+...+\cos^2(80^{\circ})+\cos^2(85^{\circ})+\cos^2(90^{\circ})}{\cos^2(90^{\circ})+\cos^2(5^{\circ})+\cos^2(10^{\circ})+...+\cos^2(85^{\circ})+1}\) Note that both the numerator and denominator have 18 terms and 1.

Calculate the ratio

Now, we will find the ratio of the number of terms in the numerator and the denominator: \(\frac{18+1}{17+1}=\frac{19}{18}\) According to the options given, we can immediately see that the answer must be (d) \(\frac{17}{19}\).

Key concepts

Trigonometric Ratios

Trigonometric ratios are the backbone of solving any trigonometry problem. These ratios, specifically sine, cosine, and tangent, connect the angles in a triangle with the lengths of its sides. Let's break down what each of these ratios means and how they are used.

- **Sine (\(\sin(x)\)):** It is the ratio of the length of the opposite side to the hypotenuse in a right triangle. For example, in a right-angled triangle, if the opposite side is 5 units and the hypotenuse is 10 units, then \(\sin(x) = \frac{5}{10} = 0.5\).

- **Cosine (\(\cos(x)\)):** This ratio is built as the length of the adjacent side divided by the hypotenuse. If the adjacent side is 8 units and the hypotenuse is 10 units, then \(\cos(x) = \frac{8}{10} = 0.8\).

- **Tangent (\(\tan(x)\)):** It is given by the ratio of the length of the opposite side to the adjacent side. So if the opposite side is 5 and the adjacent side is 8, then \(\tan(x) = \frac{5}{8}\).

Understanding these ratios helps in solving not just geometric problems, but also in analyzing waves and oscillations in physics. It's important to recognize that as angles change, these ratios represent how the ratio of the sides in a triangle change as well.

Angle Sum and Difference Identities

Angle sum and difference identities are powerful tools to solve and simplify trigonometric problems. These identities help in breaking down complex angle-related computations into simpler forms. Let's explore them.

• **Sine of Angle Sum**: \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\). This identity allows us to find the sine of an angle that can be expressed as the sum of two other angles.
• **Cosine of Angle Sum**: \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\). It similarly breaks down the cosine of a sum into a product of different sines and cosines.
• **Sine of Angle Difference**: \(\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)\) shows how we can decompose the sine of a difference.
• **Cosine of Angle Difference**: \(\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)\) simplifies problems where you have angles being subtracted.

By using these identities, any complex trigonometric expression can be handled more easily. For instance, they are especially useful when you need to calculate unknown angles or simplify equations in trigonometry.

Trigonometric Functions

Trigonometric functions are fundamental functions in mathematics, which relate the angles of a triangle to the ratios of its sides. These functions are periodic, meaning they repeat their values in regular intervals.

The three primary trigonometric functions are sine, cosine, and tangent, as previously discussed. There are also reciprocal functions such as cosecant (\(csc(x) = \frac{1}{\sin(x)}\)), secant (\(sec(x) = \frac{1}{\cos(x)}\)), and cotangent (\(cot(x) = \frac{1}{\tan(x)}\)). Each of these functions plays a vital role in solving problems involving angles and distances.

In addition, these functions have crucial properties like:

• **Periodicity**: For instance, \(\sin(x)\) and \(\cos(x)\) have a period of \(2\pi\), meaning the function values repeat every \(360^{\circ}\) or \(2\pi\) radian.
• **Even and Odd functions**: \(\cos(x)\) is an even function since \(\cos(-x) = \cos(x)\), while \(\sin(x)\) and \(\tan(x)\) are odd functions because \(\sin(-x) = -\sin(x)\) and \(\tan(-x) = -\tan(x)\).
• **Symmetry and Transformations**: \(\sin(x)\) is symmetric about the origin, while \(\cos(x)\) is symmetric about the y-axis. These properties are used to deduce the value of these functions in different quadrants of the unit circle.

Trigonometric functions are essential components in mathematics and physics and understanding their properties and behaviors is vital for mastering higher-level concepts.