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In this module you have worked with many different trigonometric identities. These identities can be helpful for making precise calculations. For example, you can use identities to find the lengths of the sides of a triangle when the angle measure in standard position is not listed on the unit circle. Most recently you have learned about double-angle and half-angle identities. These are special cases of the sum and difference formulas for sine and cosine. Consider the half-angle identities shown as you work through the scenario below. Sine Half-Angle Identity Cosine Half-Angle Identity 0 1- cose 0 1 + cose sin 2 cos = ± 2 2 Marcus is a craftsman and artist. He uses the knowledge he has gained as an engineering student to create custom pieces. For his current project, he is making a large, intricate, 3-dimensional latticework that will be placed next to the marker which welcomes people to his city. He has the work on the lattice completed. It consists of two long rectangles, such that the bottom rectangle will act as a base for the top rectangle. Together the two rectangles are 12 feet tall. 12 ft In order to help the latticework remain vertical, Marcus will attach two wires to each side of the piece, and affix them to the ground at a point that is exactly 9 feet from the base of the artwork, as shown in the illustration. The measure of the angle of elevation to the top of the base is one-half the measure of the angle of elevation to the top of the latticework. Use this information and what you know about trigonometric ratios and identities to help you explore the questions below. H+ 9 ft

In this module you have worked with many different trigonometric identities. These identities can be helpful for making precise calculations. For example, you can use identities to find the lengths of the sides of a triangle when the angle measure in standard position is not listed on the unit circle. Most recently you have learned about double-angle and half-angle identities. These are special cases of the sum and difference formulas for sine and cosine. Consider the half-angle identities shown as you work through the scenario below. Sine Half-Angle Identity Cosine Half-Angle Identity 0 1- cose 0 1 + cose sin 2 cos = ± 2 2 Marcus is a craftsman and artist. He uses the knowledge he has gained as an engineering student to create custom pieces. For his current project, he is making a large, intricate, 3-dimensional latticework that will be placed next to the marker which welcomes people to his city. He has the work on the lattice completed. It consists of two long rectangles, such that the bottom rectangle will act as a base for the top rectangle. Together the two rectangles are 12 feet tall. 12 ft In order to help the latticework remain vertical, Marcus will attach two wires to each side of the piece, and affix them to the ground at a point that is exactly 9 feet from the base of the artwork, as shown in the illustration. The measure of the angle of elevation to the top of the base is one-half the measure of the angle of elevation to the top of the latticework. Use this information and what you know about trigonometric ratios and identities to help you explore the questions below. H+ 9 ft

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Module 11, Half-Angle Identities Assignment In this module you have worked with many different trigonometric identities. These identities can be helpful for making precise calculations. For example, you can use identities to find the lengths of the sides of a triangle when the angle measure in standard position is not listed on the unit circle. Most recently you have learned about double-angle and half-angle identities. These are special cases of the sum and difference formulas for sine and cosine. Consider the half-angle identities shown as you work through the scenario below. Sine Half-Angle Identity Cosine Half-Angle Identity 1- cose 1+ cose cos = + sin = + 2 2 Marcus is a craftsman and artist. He uses the knowledge he has gained as an engineering student to create custom pieces. For his current project, he is making a large, intricate, 3-dimensional latticework that will be placed next to the marker which welcomes people to his city. He has the work on the lattice completed. It consists of two long rectangles, such that the bottom rectangle will act as a base for the top rectangle. Together the two rectangles are 12 feet tall. In order to help the latticework remain vertical, Marcus will attach two wires to each side of the piece, and affix them to the ground at a point that is exactly 9 feet from the 12 ft base of the artwork, as shown in the illustration. The measure of the angle of elevation to the top of the base is one-half the measure of the angle of elevation to the top of the latticework. Use this information and what you know about trigonometric ratios and identities to help you explore the questions below. 9 ft
Transcribed Image Text:Module 11, Half-Angle Identities Assignment In this module you have worked with many different trigonometric identities. These identities can be helpful for making precise calculations. For example, you can use identities to find the lengths of the sides of a triangle when the angle measure in standard position is not listed on the unit circle. Most recently you have learned about double-angle and half-angle identities. These are special cases of the sum and difference formulas for sine and cosine. Consider the half-angle identities shown as you work through the scenario below. Sine Half-Angle Identity Cosine Half-Angle Identity 1- cose 1+ cose cos = + sin = + 2 2 Marcus is a craftsman and artist. He uses the knowledge he has gained as an engineering student to create custom pieces. For his current project, he is making a large, intricate, 3-dimensional latticework that will be placed next to the marker which welcomes people to his city. He has the work on the lattice completed. It consists of two long rectangles, such that the bottom rectangle will act as a base for the top rectangle. Together the two rectangles are 12 feet tall. In order to help the latticework remain vertical, Marcus will attach two wires to each side of the piece, and affix them to the ground at a point that is exactly 9 feet from the 12 ft base of the artwork, as shown in the illustration. The measure of the angle of elevation to the top of the base is one-half the measure of the angle of elevation to the top of the latticework. Use this information and what you know about trigonometric ratios and identities to help you explore the questions below. 9 ft
1. Can you use the given information to calculate the measure of the angle of elevation to the top of the latticework? If so, find the angle measure to the nearest degree. Show your work. If not, explain why not and what information is needed. 2. What length of wire will Marcus need to attach the top of the latticework to the ground? Show your work. 3. Can you use the given information to find the exact height of the base? If so, find the height and show your work. If not, explain why not and what information is needed. 4. Consider your responses to question 3. Devise a plan to find any information that you need in order to find the height of the base.
Transcribed Image Text:1. Can you use the given information to calculate the measure of the angle of elevation to the top of the latticework? If so, find the angle measure to the nearest degree. Show your work. If not, explain why not and what information is needed. 2. What length of wire will Marcus need to attach the top of the latticework to the ground? Show your work. 3. Can you use the given information to find the exact height of the base? If so, find the height and show your work. If not, explain why not and what information is needed. 4. Consider your responses to question 3. Devise a plan to find any information that you need in order to find the height of the base.
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