Problem 18 - Precalculus with Limits

Question

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=60^{\circ}, \quad a=9, \quad c=10$$

Step-by-Step Solution

Answer

Using the Law of Sines, the angles and sides of the triangle have been calculated as such: B = 180° - A - C, b = a sin(B) / sin(A), and sin C = c sin(A) / a. The precise values would be determined by using a scientific calculator to perform the trigonometric calculations.

Step 1: Define the Law of Sines

The Law of Sines is defined as the ratio of the length of a side of a triangle to the sine of its opposite angle being constant. This can be written as: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), where a, b, and c are sides and A, B, and C are the angles of the triangle.

Step 2: Apply the Law of Sines to Find Angle C

From the given triangle, we have A = 60°, a = 9, and c = 10. Applying the law of sines, we find that \(\sin C = \frac{c \sin A}{a} = \frac{10 \sin 60°}{9}\), which can be simplified to get C.

Step 3: Find the Missing Angle

The sum of angles in a triangle adds up to 180°. So, by subtracting the known angles A and C from 180°, the angle B can be found. Thus, B = 180° - A - C.

Step 4: Apply the Law of Sines Again to Find Side b

According to what we have found, we can use the law of sines again to find side b using the formula: \(b = a \sin B / \sin A = 9 \sin B / \sin 60°\), which can be simplified to get side b.

Key Concepts

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between angles and sides of triangles. It's a tool that helps us solve various geometric problems by providing formulas that relate sides and angles to each other. In a right triangle, the primary trigonometric functions—sine, cosine, and tangent—are used frequently. However, these functions are not just limited to right triangles. They extend to any triangle, allowing us to find unknown side lengths and angles. The Law of Sines is a key component of trigonometry that applies to any type of triangle. It helps us find missing values by expressing relationships between side lengths and corresponding angles, adding power to the problem-solving toolkit in trigonometry.

Triangle Solving

Solving a triangle means finding all the unknown sides and angles when certain measurements are given. Thanks to the properties of triangles, we have a lot of tools at our disposal. Notably, any triangle's angles sum up to 180°.

• To start solving, gather all known side lengths and angles.
• Use formulas like the Law of Sines to find unknown angles if some angles and their opposite side lengths are known.
• Calculate the remaining side lengths once all the angles are known.

By implementing these strategies, you can successfully solve most problems related to triangles, ensuring that you have a comprehensive understanding of their properties.

Angle Calculation

Calculating angles is a fundamental part of solving triangles, which involves determining the size of angles when certain sides or angles are known. The Law of Sines is particularly useful in finding angles when at least one opposite side length and angle are known.For instance, in our exercise, to calculate angle C, we used: \[sin C = \frac{c \sin A}{a}\]Subsequently, you can find another angle using the angle sum property of triangles, where the sum of all angles in a triangle should equal 180°. If two angles are known, simply subtract them from 180° to find the remaining angle. Such calculations are vital in obtaining a complete picture of the triangle's geometry.

Sine Function

The sine function is one of the basic trigonometric functions used to relate the angles of a triangle to the lengths of the sides opposite those angles. In terms of a right triangle, it is the ratio of the length of the opposite side to the hypotenuse. In solving non-right triangles, like in our example, the sine function is pivotal as part of the Law of Sines. It allows us to equate ratios of side lengths and sine values of their corresponding opposite angles, thus finding unknown values. This versatility exemplifies how the sine function extends beyond simple angle calculations and becomes an essential part of solving triangles in trigonometry.