OVERVIEW Similarity Leads to Trigonometry G.SRT.6 G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Develop the foundational concepts of trigonometry from similarity. helps us know that all 30 degree right triangles are similar and so they will have proportional sides and fied ratios. This leads nicely into the process to establish Sine, osine and Tangent for all right triangles. Trigonometry is similarity. Proportional sides of similar triangles provide equivalent ratios of those sides. () The student will be able to label a triangle in relation to the reference angle (opposite, adjacent & hypotenuse). () The student will be able to determine the most appropriate trigonometric ratio (sine, cosine, and tangent) to use for a given problem based on the information provided. (3) The student will be able to solve for sides and angles of right triangles using trigonometry. This objective is really getting at the conceptual understanding of trigonometry and how it is established from similarity and proportional sides. Trigonometry is a major topic and probably one of the most used pieces of mathematics outside of the classroom. It has far reaching uses because of its ability to relate angles to sides. Use the special right triangle relationships to develop the Sine, osine and Tangent values and relate them to a trigonometry table. Introduce trigonometry using a table first and then a calculator. The table allows students to see number patterns and develop a more conceptual understanding of what is going on. gsrt6similarityleadstotrigonometry Page of 6 4/9/04
NOTES Similarity Leads to Trigonometry G.SRT.6 ONEPT Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. In G.SRT.5 it was discussed that all - - 90 right isosceles triangles have the same relationships of its sides no matter how large or small the triangle is. These ratio relationships are unchanging due to the similarity of the triangles. Similarity preserves the angle measures and maintains the sides proportionality. 7 cm 7 cm 7 cm 5 cm 5 cm 5 cm 9 6 cm 9 3 cm 9 3 cm It is this proportionality of all 45-45 - 90 that allows for the generalization of these relationships. The ratios within the triangle, such as the leg to the hypotenuse or the leg to the leg, are fied values because they would remain proportional no matter the size of the triangle. alculating them, it is found that: The ratio of leg to hypotenuse: leg 0.707 hypotenuse This will be true for LL 45-45 -90 triangles. The ratio of leg to leg: leg leg This will be true for LL 45-45 -90 triangles. When looking at ratios within the triangle name the sides so that there is no confusion on which side or leg that is being referred to. When the sides are labeled it will always be based on a reference angle. In the previous eample there was no need to discuss this because the two reference angles were the same. o djacent Leg Leg Hypotenuse If is the reference angle, then is the opposite leg, is the adjacent leg, and is the hypotenuse. The hypotenuse will always be opposite the right angle (the longest side). The opposite leg will always be the side of the triangle that does not form the reference angle. The adjacent leg will always be the non-hypotenuse side that forms the reference angle. o Leg djacent Leg Hypotenuse If is the reference angle, then is the opposite leg, is the adjacent leg, and is the hypotenuse. gsrt6similarityleadstotrigonometry Page of 6 4/9/04
NOTES Similarity Leads to Trigonometry G.SRT.6 TRIGONOMETRY (Triangle Measure) allows the linkage between angle size in right triangles to side proportionality. If the reference angle in a right triangle is known, then there are three fied ratios of its sides. The converse is also true, if the ratio of two sides of a right triangle is known then there eists one reference angle that has that ratio. Trigonometry connects angle measure to side measure, thus it is a very powerful piece of mathematics. The three ratios established have names: The Sine Ratio (sin) The osine Ratio (cos) The Tangent Ratio (tan) djacent sin cos tan Hypotenuse Hypotenuse djacent Lets look at how trigonometry helps in the discovery of information about triangles. portion of the table has been clipped to demonstrate the power of these numbers. Knowing Sides. Get ngles Knowing ngles Get Sides θ.0 cm (adjacent) tan djacent.6 tan.9.0 tan.9.6 cm (opposite) The angle must be approimately 50 because its tangent ratio is.9. opposite Without knowing anything about the opposite and the hypotenuse other than the reference angle, the value of ratio between them is known. sin 54 opposite hypotenuse hypotenuse 54 opposite hypotenuse 0.8090 Degrees Sine (sin) osine (cos) Tangent (tan) 46 0.793 0.6947.0355 47 0.734 0.680.074 48 0.743 0.669.06 49 0.7547 0.656.504 50 0.7660 0.648.98 5 0.777 0.693.349 5 0.7880 0.657.799 53 0.7986 0.608.370 54 0.8090 0.5878.3764 The chart allows users to find ratios of sides if angle measures are known or to find the reference angle if the ratio of sides is given. The chart is a nice way to learn the concept but calculators are much more powerful because they can handle angle sizes other than whole numbers. gsrt6similarityleadstotrigonometry Page 3 of 6 4/9/04
SSESSMENT Similarity Leads to Trigonometry G.SRT.6. The opposite side of is: ) ) ) D) Depends. The Sine ratio (opposite/hypotenuse) in a 30 right triangle is: ) 3 ) ) D) 3 3. The Sine ratio (opposite/hypotenuse) in a 45 right triangle is: ) ) ) D) 4. If is the reference angle, then the tangent ratio would be: ) 5 7 5. The ratio 5 3 ) 8 5 represent the which relationship: ) 8 7 D) 5 8 ) sin ) tan ) tan D) cos 7 cm 8 cm 5 cm cm 5 cm 3 cm 6. The value of Ɵ is approimately: ) 8.43 ) 9.47 ) 65.4 D) 70.53 5 cm 5 cm 7. The value of is approimately: 30 Degrees Sine (sin) osine (cos) Tangent (tan) 0 cm 30 0.5000 0.8660 0.5774 ) 5.000 cm ) 5.774 cm 40 0.648 0.7660 0.839 50 0.7660 0.648.98 ) 6.48 cm D) 8.660 cm gsrt6similarityleadstotrigonometry Page 4 of 6 4/9/04
SSESSMENT Similarity Leads to Trigonometry G.SRT.6 8. The value of Ɵ is approimately: 0 cm 4.5 cm Degrees Sine (sin) osine (cos) Tangent (tan) 5 0.46 0.9063 0.4663 ) 5 ) 35 35 0.5736 0.89 0.700 45 0.707 0.707.0000 55 0.89 0.5736.48 ) 45 D) 55 9. Jeremy looks at the given triangle and says The measure of angle is 30. How could Jeremy determine this from only measurements on diagram and not any other tools? 0 cm 5 cm 0. When looking at a trigonometry table Sarah notices that the Sine and osine ratios are the same value for the 45 right triangle. Eplain why that happened.. When looking at the trigonometry table Jessica notices that the Sine ratio for all angles between 0 and 90 are values between 0 and. Why is this an appropriate range of values for the Sine ratio?. Solve for the missing information. (Round all final answers to decimals places) a) b) c) d) cm 36 8 cm cm 63 0 cm 9 cm 3 cm Ɵ Ɵ e) f) g) h) 58 cm 9 cm 3 cm 4 cm cm 4 cm Ɵ Ɵ gsrt6similarityleadstotrigonometry Page 5 of 6 4/9/04
SSESSMENT Similarity Leads to Trigonometry G.SRT.6 nswers: ) ) 3) 4) 5) D 6) D 7) 8) 9) The ONLY right triangle that has a ratio of opposite : hypotenuse that is : is a 30-60 - 90. Thus m = 30 because then the ratio of opposite : hypotenuse is :. 0) 45 right triangle is an isosceles triangle with congruent legs. These two congruent legs are the opposite and the adjacent for the 45 angle so opposite hypotenuse is the same as adjacent hypotenuse. ) The sine ratio is the comparison of the opposite. The hypotenuse is the longest side, so hypotenuse when we compare an opposite side to the hypotenuse we will always have value between 0 and. ) a) 8.7 cm b) 43.34 c) 0.0 cm d) 73.30 e) 4.4 f) 5.33 cm g) 5.85 cm h) 59.00 gsrt6similarityleadstotrigonometry Page 6 of 6 4/9/04