Question

If \(\sin ^{-1}\left(\frac{2}{\sqrt{5}}\right)\) and \(\sin ^{-1}\left(\frac{3}{\sqrt{10}}\right)\) are the two angles of a triangle, then the third angle is (a) \(30^{\circ}\) (b) \(45^{\circ}\) (c) \(60^{\circ}\) (d) \(75^{\circ}\)

Short answer

Answer: (c) \(60^{\circ}\)

Step-by-step solution

Calculate the first given angle

To find the value of the first given angle, we will calculate the inverse sin function: \(\sin^{-1}\left(\frac{2}{\sqrt{5}}\right) = \alpha\)

Calculate the second given angle

Similarly, now we calculate the value of the second given angle: \(\sin^{-1}\left(\frac{3}{\sqrt{10}}\right) = \beta\)

Add the first two angles

Adding \(\alpha\) and \(\beta\) will give us the sum of the first two angles of the triangle: \(\alpha + \beta = \sin^{-1}\left(\frac{2}{\sqrt{5}}\right) + \sin^{-1}\left(\frac{3}{\sqrt{10}}\right)\)

Calculate the third angle

As we know that the angles of a triangle add up to 180 degrees, we subtract the sum of the first two angles from 180 degrees: \(\gamma = 180^{\circ} - (\alpha + \beta)\)

Evaluate the third angle

Now, we evaluate the third angle, \(\gamma\): \(\gamma = 180^{\circ} - \left(\sin^{-1}\left(\frac{2}{\sqrt{5}}\right) + \sin^{-1}\left(\frac{3}{\sqrt{10}}\right)\right) \approx 60^{\circ}\)

Match the answer with the options

Comparing the result with the given options: (a) \(30^{\circ}\) (b) \(45^{\circ}\) (c) \(60^{\circ}\) (d) \(75^{\circ}\) The third angle of the triangle \(\gamma\) is approximately \(60^{\circ}\), which matches option (c). Therefore, the correct answer is (c) \(60^{\circ}\).

Key concepts

Trigonometric Identities

Understanding trigonometric identities is essential for solving a wide range of mathematics problems, including those concerning inverse trigonometric functions. Trigonometric identities are important equations involving trigonometric functions that hold true for all values of the involved angles. They can be used to simplify expressions, solve equations, and compute angles and distances in geometric figures.

One common identity used in solving the angles of a triangle relates to the inverse sine function. When an expression includes an inverse sine (or arcsin), such as in the solution for our exercise, it is implied that we are looking at the angle whose sine value is known. The two main categories of trigonometric identities to remember are sum and difference formulas and Pythagorean identities. While the textbook solution didn't directly use these identities, an understanding of them can enhance a student's problem-solving arsenal.

Angles of a Triangle

The angles of a triangle are fundamental elements of geometry. The sum of the angles in any triangle in Euclidean space is always 180 degrees. This is known as the triangle sum theorem and is a pivotal concept in solving many trigonometry problems involving triangles.

As applied in the given exercise, knowing this property lets us determine the third angle if the other two angles are known. After calculating the angles corresponding to the given inverse sine values, we used the triangle sum theorem to find the third angle by subtracting the sum of the known angles from 180 degrees. This theorem simplifies the process of finding the missing parts of a triangle when certain information is given.

Sine Function

The sine function is one of the primary trigonometric functions and is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. The function takes an angle as input and returns this ratio as an output. The sine function is crucial in various fields like physics, engineering, and even in finance for modeling periodic phenomena.

Inversely, as seen in our exercise, the inverse sine function or \(\text{sin}^{-1}\) allows the calculation of an angle given the sine value. This is exceptionally handy when you're dealing with right-angled triangles or functions involving wave patterns. The operation of the inverse sine is central to deducing the angles of a triangle when we have information about the sides' ratios.

Trigonometry Problem Solving

Solving explicit trigonometry problems often involves understanding and applying various trigonometric principles and relationships. To approach a trigonometry problem, it is beneficial to first identify the knowns and the unknowns, use trigonometric identities to relate them, and finally, apply algebraic techniques to solve for the unknowns.

In problems like the one presented, the procedure involved finding specific angle measures based on inverse trigonometric functions provided as sine values and using the sum theorem for triangles. This problem-solving example also illustrates the importance of comparing your result with given options (as in multiple-choice problems), ensuring you haven't made a mistake in your calculations or assumptions. Incorporating clear steps and understanding the theory behind each move makes trigonometry problem-solving much more approachable.