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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 11
Chapter 8, Problem 11

Find each angle B. Do not use a calculator.


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Verified step by step guidance
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Identify the given information from the problem, such as the sides or angles related to angle B. Since the image is not provided, assume you have either side lengths or trigonometric ratios involving angle B.
Recall the primary trigonometric ratios: sine, cosine, and tangent, which relate the angles of a right triangle to the ratios of its sides. For example, \( \sin B = \frac{\text{opposite}}{\text{hypotenuse}} \), \( \cos B = \frac{\text{adjacent}}{\text{hypotenuse}} \), and \( \tan B = \frac{\text{opposite}}{\text{adjacent}} \).
Choose the appropriate trigonometric ratio based on the sides or values given in the problem that involve angle B. Write the equation for that ratio using the known values.
To find angle B, use the inverse trigonometric function corresponding to the ratio you set up. For example, if you used sine, then \( B = \sin^{-1}(\text{value}) \); if cosine, then \( B = \cos^{-1}(\text{value}) \); if tangent, then \( B = \tan^{-1}(\text{value}) \).
Since the problem states not to use a calculator, use known special angle values or trigonometric identities to determine angle B from the ratio. For example, recognize common ratios like \( \frac{1}{2} \), \( \frac{\sqrt{3}}{2} \), or 1, which correspond to angles like 30°, 45°, or 60°.

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Key Concepts

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

Basic Trigonometric Ratios

Understanding sine, cosine, and tangent ratios is essential for finding unknown angles in right triangles. These ratios relate the angles to the lengths of the sides, allowing calculation of angles when side lengths are known.
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Introduction to Trigonometric Functions

Inverse Trigonometric Functions

Inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) are used to find angles from given ratio values. Since the question prohibits calculators, recognizing common angle values and their ratios is important.
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Introduction to Inverse Trig Functions

Special Right Triangles

Knowledge of special right triangles (30°-60°-90° and 45°-45°-90°) helps identify angles without a calculator by recalling their side ratios. This aids in quickly determining angle B when side lengths match these patterns.
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45-45-90 Triangles
Related Practice
Textbook Question

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

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Textbook Question

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

〈-4, -7〉

6
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Textbook Question

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

〈15, -8〉

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 magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary. See Example 1.

〈-7, 24〉

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