Trigonometry and Modelling Fundamentals
Explore essential trigonometry concepts and modeling techniques. Learn to apply addition formulae, solve equations, and prove trigonometric relationships using diagrams and known properties. Enhance your understanding of sine, cosine, angles, and hypotenuse in various scenarios.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Trigonometry and Trigonometry and Modelling Modelling Twitter: @Owen134866 www.mathsfreeresourcelibrary.com
Prior Knowledge Check 3) Prove the following: 1) Find the exact values of: a) ???? + ???????? ???? b) ???? 3 1 2 a) ???45 2 or 2 6 2 c) ???? 3 3 b) ???????????? 1 2) Solve the following equations in the interval 0 ? 360 c) ???2?+???2? 1+???2? ???2? a) sin ? + 50 = 0.9 b) cos 2? 30 =1 c) 2???2? ???? 3 = 0 194.2,245.8 45,165,225,345 2 270
Teachings for Teachings for Exercise 7A Exercise 7A
Trigonometry and Modelling ? ? You need to be able to use the addition formulae for ????, ?????? and ??? ? 1 90 ? ? ? ? In the diagram to the right, ??? = ?, ??? = ?, and ?? = 1. Additionally, lines ?? and ?? are perpendicular, lines ?? and ?? are perpendicular, and lines ?? and ?? are perpendicular. ? ? ? ? ? Since ?? and ?? are parallel, ??? = ? This means that ??? = 90 ? Use the diagram, together with known properties of sine and cosine, to prove the following: Which then means that ??? = ? ??? ? + ? = ???????? + ???????? 7A
Trigonometry and Modelling ? ? You need to be able to use the addition formulae for ????, ?????? and ??? ? ???? 1 ? ? In the diagram to the right, ??? = ?, ??? = ?, and ?? = 1. Additionally, lines ?? and ?? are perpendicular, lines ?? and ?? are perpendicular, and lines ?? and ?? are perpendicular. ? ? ? ? ? In a right-angled triangle: ??? = ???? ??? In triangle ???, the angle is ? and the Hypotenuse is 1 Use the diagram, together with known properties of sine and cosine, to prove the following: ??? = ???? 1 ??? = ???? ??? ? + ? = ???????? + ???????? 7A
Trigonometry and Modelling ? ? You need to be able to use the addition formulae for ????, ?????? and ??? ? ???????? ???? 1 ? ? In the diagram to the right, ??? = ?, ??? = ?, and ?? = 1. Additionally, lines ?? and ?? are perpendicular, lines ?? and ?? are perpendicular, and lines ?? and ?? are perpendicular. ? ? ? ? ? In a right-angled triangle: ??? = ???? ??? In triangle ???, the angle is ? and the Hypotenuse is ???? Use the diagram, together with known properties of sine and cosine, to prove the following: ??? = ???? ???? ??? = ???????? ??? ? + ? = ???????? + ???????? 7A
Trigonometry and Modelling ? ? You need to be able to use the addition formulae for ????, ?????? and ??? ? ???????? ???? ???????? 1 ? ? In the diagram to the right, ??? = ?, ??? = ?, and ?? = 1. Additionally, lines ?? and ?? are perpendicular, lines ?? and ?? are perpendicular, and lines ?? and ?? are perpendicular. ? ? ? ? ? In a right-angled triangle: ??? = ???? ??? In triangle ???, the angle is ? and the Hypotenuse is ???? Use the diagram, together with known properties of sine and cosine, to prove the following: ??? = ???? ???? ??? = ???????? ??? ? + ? = ???????? + ???????? 7A
Trigonometry and Modelling ? ? You need to be able to use the addition formulae for ????, ?????? and ??? ? ???????? ???? ???????? 1 ? ? In the diagram to the right, ??? = ?, ??? = ?, and ?? = 1. Additionally, lines ?? and ?? are perpendicular, lines ?? and ?? are perpendicular, and lines ?? and ?? are perpendicular. ? ? ? ? ? In a right-angled triangle: ??? = ???? ??? In triangle ???, the angle is ? and the Hypotenuse is 1 Use the diagram, together with known properties of sine and cosine, to prove the following: ??? = ???? 1 ??? = ???? ??? ? + ? = ???????? + ???????? 7A
Trigonometry and Modelling ? ? You need to be able to use the addition formulae for ????, ?????? and ??? ? ???????? ???? ???????? 1 ? ? In the diagram to the right, ??? = ?, ??? = ?, and ?? = 1. Additionally, lines ?? and ?? are perpendicular, lines ?? and ?? are perpendicular, and lines ?? and ?? are perpendicular. ???????? ? ? ? ? ? In a right-angled triangle: ??? = ???? ??? In triangle ???, the angle is ? and the Hypotenuse is ???? Use the diagram, together with known properties of sine and cosine, to prove the following: ??? = ???? ???? ??? = ???????? ??? ? + ? = ???????? + ???????? 7A
Trigonometry and Modelling ? ? You need to be able to use the addition formulae for ????, ?????? and ??? ? ???????? ???? ???????? 1 ? ? In the diagram to the right, ??? = ?, ??? = ?, and ?? = 1. Additionally, lines ?? and ?? are perpendicular, lines ?? and ?? are perpendicular, and lines ?? and ?? are perpendicular. ???????? ? ? ? ? ? ???????? In a right-angled triangle: ??? = ???? ??? Use the diagram, together with known properties of sine and cosine, to prove the following: In triangle ???, the angle is ? and the Hypotenuse is ???? ??? = ???? ???? ??? ? + ? = ???????? + ???????? ??? = ???????? 7A
Trigonometry and Modelling ? ? You need to be able to use the addition formulae for ????, ?????? and ??? ? ???????? ???? ???????? 1 ? ? In the diagram to the right, ??? = ?, ??? = ?, and ?? = 1. Additionally, lines ?? and ?? are perpendicular, lines ?? and ?? are perpendicular, and lines ?? and ?? are perpendicular. ???????? ? ? ? ? ? ???????? In a right-angled triangle: ??? = ???? ??? Use the diagram, together with known properties of sine and cosine, to prove the following: In triangle ???, the angle is (? + ?) and the Hypotenuse is 1 ??? = ???(? + ?) 1 ??? = ???(? + ?) ??? ? + ? = ???????? + ???????? However, we also know that the Opposite side is ???????? + ???????? ??? = ???????? + ???????? 7A
??? ? + ? = ???????? + ???????? Trigonometry and Modelling ? ? You need to be able to use the addition formulae for ????, ?????? and ??? ? ???????? ???? ???????? 1 ? ? In the diagram to the right, ??? = ?, ??? = ?, and ?? = 1. Additionally, lines ?? and ?? are perpendicular, lines ?? and ?? are perpendicular, and lines ?? and ?? are perpendicular. ???????? ? ? ? ? ? ???????? ??? = ???(? + ?) ??? = ???????? + ???????? Use the diagram, together with known properties of sine and cosine, to prove the following: Therefore ??? ? + ? = ???????? + ???????? ??? ? + ? = ???????? + ???????? 7A
??? ? + ? = ???????? + ???????? Trigonometry and Modelling ? ? You need to be able to use the addition formulae for ????, ?????? and ??? ? ???????? ???? ???????? 1 ? ? In the diagram to the right, ??? = ?, ??? = ?, and ?? = 1. Additionally, lines ?? and ?? are perpendicular, lines ?? and ?? are perpendicular, and lines ?? and ?? are perpendicular. ???????? ? ? ? ? ? ???????? In a right-angled triangle: ??? = ???? ??? Use the diagram, together with known properties of sine and cosine, to prove the following: In triangle ???, the angle is (? + ?) and the Hypotenuse is 1 ??? = ???(? + ?) 1 ??? = ???(? + ?) ??? ? + ? = ???????? ???????? However, we also know that the Adjacent side is ???????? ???????? ??? = ???????? ???????? 7A
??? ? + ? = ???????? + ???????? ??? ? + ? = ???????? ???????? Trigonometry and Modelling ? ? You need to be able to use the addition formulae for ????, ?????? and ??? ? ???????? ???? ???????? 1 ? ? In the diagram to the right, ??? = ?, ??? = ?, and ?? = 1. Additionally, lines ?? and ?? are perpendicular, lines ?? and ?? are perpendicular, and lines ?? and ?? are perpendicular. ???????? ? ? ? ? ? ???????? ??? = ???(? + ?) ??? = ???????? ???????? Use the diagram, together with known properties of sine and cosine, to prove the following: Therefore ??? ? + ? = ???????? ???????? ??? ? + ? = ???????? ???????? 7A
??? ? + ? = ???????? + ???????? ??? ? ? = ???????? ???????? ??? ? + ? = ???????? ???????? ??? ? ? = ???????? + ???????? Trigonometry and Modelling You need to be able to use the addition formulae for ????, ?????? and ??? ??? ? + ? = ???????? ???????? Replace ? and ? with ? and ? ??? ? + ? = ???????? ???????? Replace ? with ( ?) You can use some trig relationships Use the results above to show that: ??? ? + ( ?) = ???????( ?) ???????( ?) ??? ? ? = ???????? ????( ????) ??? ? ? = ???????? + ???????? Simplify ??? ? ? = ???????? + ???????? ???? = ???( ?) For A similar process can be followed for ??? ? ? ! example ???20 = ???( 20) ???? = ???? For example ???40 = ???40 7A
???? + ???? 1 ???????? ??? ? + ? = ???????? + ???????? ??? ? ? = ???????? ???????? ??? ? + ? = ???????? ???????? ??? ? + ? = ??? ? ? = ???????? + ???????? Trigonometry and Modelling ???? ???? 1 + ???????? ??? ? ? = ??? ? + ? =??? ? + ? You need to be able to use the addition formulae for ????, ?????? and ??? ??? ? + ? Replace using the relationships above ??? ? + ? =???????? + ???????? ???????? ???????? Use the results above to show that: Divide all terms by ???????? ???????? ????????+???????? ???????? ???????? ???????? ???????? ??? ? + ? = ???? + ???? 1 ???????? ???????? Simplify each term ??? ? + ? = ???? + ???? 1 ???????? ??? ? + ? = A similar process can be followed for ??? ? ? ! 7A
???? + ???? 1 ???????? ??? ? + ? = ???????? + ???????? ??? ? ? = ???????? ???????? ??? ? + ? = ???????? ???????? ??? ? + ? = ??? ? ? = ???????? + ???????? Trigonometry and Modelling ???? ???? 1 + ???????? ??? ? ? = You need to be able to use the addition formulae for ????, ?????? and ??? ???? ???? ???? ???? Multiply to make the denominators equal Prove that: =???????? ???????? ???????? ???????? Group together ???? ???? ???? ???? ??? ? + ? ???????? =???????? ???????? ???????? The top can be replaced using one of the identities above =???(? + ?) ???????? 7A
???? + ???? 1 ???????? ??? ? + ? = ???????? + ???????? ??? ? ? = ???????? ???????? ??? ? + ? = ???????? ???????? ??? ? + ? = ??? ? ? = ???????? + ???????? Trigonometry and Modelling ???? ???? 1 + ???????? ??? ? ? = Given that: 2??? ? + ? = 3???(? ?) Express ???? in terms of ???? 2??? ? + ? = 3???(? ?) Rewrite the sin and cos parts 2(???????? + ????????) = 3(???????? + ????????) Multiply out the brackets 2???????? + 2???????? = 3???????? + 3???????? Divide all by cosxcosy 2???????? + 2???????? = 3???????? + 3???????? ???????? ???????? ???????? ???????? Simplify 2???? + 2???? = 3+3???????? Subtract 3tanxtany Subtract 2tany 2???? 3???????? = 3 2???? Factorise the left side ????(2 3????) = 3 2???? Divide by (2 3tany) ???? = 3 2???? 2 3???? 7A
