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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblem 9
Chapter 8, Problem 9

Find the length of the remaining side of each triangle. Do not use a calculator.


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Identify the type of triangle and the given sides or angles from the image. Typically, you will have either a right triangle or a non-right triangle with some sides and angles labeled.
If the triangle is a right triangle, use the Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\), \(b\) are the legs. Rearrange the formula to solve for the missing side.
If the triangle is not a right triangle but you know two sides and the included angle, use the Law of Cosines: \(c^2 = a^2 + b^2 - 2ab \cos(C)\), where \(C\) is the angle opposite side \(c\). Substitute the known values and solve for the missing side.
If you know two angles and one side, use the Law of Sines: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). Use this to find the missing side by setting up a proportion with the known side and angle.
After setting up the appropriate equation, simplify and solve algebraically for the missing side length. Since no calculator is allowed, use exact values of sine or cosine for common angles (like 30°, 45°, 60°) or simplify the square roots as needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Pythagorean Theorem

The Pythagorean theorem relates the lengths of the sides in a right triangle: the square of the hypotenuse equals the sum of the squares of the other two sides. It is essential for finding a missing side length when two sides are known.
Recommended video:
5:19
Solving Right Triangles with the Pythagorean Theorem

Right Triangle Properties

Understanding that the triangle is right-angled allows the use of specific relationships like the Pythagorean theorem and trigonometric ratios. Recognizing the right angle is crucial for applying these methods correctly.
Recommended video:
5:35
30-60-90 Triangles

Exact Values and Simplification

Since the problem specifies not to use a calculator, it is important to work with exact values and simplify square roots or expressions manually. This skill helps in expressing the answer in simplest radical form.
Recommended video:
5:08
Sine, Cosine, & Tangent of 30°, 45°, & 60°
Related Practice
Textbook Question

Determine the number of triangles ABC possible with the given parts.


c = 50, b = 61, C = 58°

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Textbook Question

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.

〈5, 7〉

Textbook Question

Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.


a + b

Textbook Question

Consider each case and determine whether there is sufficient information to solve the triangle using the law of sines.


Three sides are known.

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Textbook Question

Find the unknown angles in triangle ABC for each triangle that exists.


A = 142.13°, b = 5.432 ft, a = 7.297 ft

Textbook Question

Solve each triangle ABC that exists.

A = 96.80°, b = 3.589 ft, a = 5.818 ft