Grade 10 Academic (MPM2D) Unit 6: Trigonometry 2: Non-Right Triangles Sine Law Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved

Trigonometry is a branch of mathematics that studies the relationship between the measures of the angles and the lengths of the sides in triangles. Sine Law © 2017 E. Choi – MPM2D - All Rights Reserved

Notation: In any triangle, the vertices are named using capital letters as in the given example. The lengths of the sides are named using lower case letters that match the opposite vertex. Sine Law © 2017 E. Choi – MPM2D - All Rights Reserved

Sine (Sin), COSINE (Cos), TANGENT (Tan): Opposite side Adjacent side Hypotenuse side Sine Law © 2017 E. Choi – MPM2D - All Rights Reserved

Earlier you explored the relations between the angle measures and sides of right triangles. You developed ratios of the lengths of the sides of similar triangles and called these trigonometric ratios sine, cosine, and tangent. Of course, not all real-world situations can be modelled with right triangles. Other methods must be found to solve problems that involve general, non-right triangles. Sine Law © 2017 E. Choi – MPM2D - All Rights Reserved

Investigations Given , complete the chart below What do you notice? The trigonometric ratios between the angle and corresponding side are same. 80o 65o 35o 8.7 cm 8 cm 5 cm 0.11 0.11 0.11 Accurate to 2 decimal places Sine Law © 2017 E. Choi – MPM2D - All Rights Reserved

Non-right triangles Sine Law Trigonometric Ratios are no longer work for non-right triangles Instead: Sine law and Cosine law Sine Law or Similar idea for A Sine Law © 2017 E. Choi – MPM2D - All Rights Reserved

Example 1: Sine Law Given the diagram of ABC, A = 71. 2°, B = 63 Example 1: Sine Law Given the diagram of ABC, A = 71.2°, B = 63.7°, and a = 75cm. Solve the triangle. 71.2° b c Supplementary Angles Sine Law 63.7° 75 cm Reference Ratio Sine Law © 2017 E. Choi – MPM2D - All Rights Reserved

Example 2: Given the diagram of DEF, F = 35°, f = 42 m, and d = 64 m Example 2: Given the diagram of DEF, F = 35°, f = 42 m, and d = 64 m. Solve the triangle. (1 decimal place for side, nearest degree for angles) Sine Law e Reference Ratio 42 m 35° 64 cm Supplementary Angles © 2017 E. Choi – MPM2D - All Rights Reserved Sine Law

Sine Law Supplementary Angles Therefore total fencing needed: Example 3: The town surveyor has to stake the lot market for a new public park beside an existing building lot. The engineering department gave this sketch. How much chain-link fence will be needed to enclose the entire park? Supplementary Angles Sine Law Reference Ratio Therefore total fencing needed: = 46.0m + 36.3m + 45.0m = 127.3m Sine Law © 2017 E. Choi – MPM2D - All Rights Reserved

Homework Work sheet: Day 1: Sine Law Text: Day 2: P. 549 #1-4, 6-8 P. 550 #10, 15 P. 555 #1, 5, 9, 12 Check the website for updates Sine Law © 2017 E. Choi – MPM2D - All Rights Reserved

End of lesson Sine Law © 2017 E. Choi – MPM2D - All Rights Reserved