Find the Exact Values of the Three Trignometric Functions of the Angle

question 36

Multiple Choice

Find the exact values of the three trignometric functions of the angle $\theta$ $( \sin \theta , \cos \theta , \tan \theta )$ of the two triangles shown below. Is the function values are the same? Explain.

• A) \(\sin \theta _ { 1 } = \frac { 5 } { 13 } , \cos \theta _ { 1 } = \frac { 12 } { 13 } , \tan \theta _ { 1 } = \frac { 5 } { 12 }\) \(\sin \theta = \frac { 6.5 } { 2.5 } = \frac { 13 } { 5 } , \cos \theta = \frac { 6.5 } { 6 } = \frac { 13 } { 12 } , \tan \theta = \frac { 6 } { 2.5 } = \frac { 12 } { 5 }\) The function values are different because the traingles are not similar and corresponding sides are not proportional.
• B) \(\sin \theta = \frac { 12 } { 13 } , \cos \theta = \frac { 5 } { 13 } , \tan \theta = \frac { 5 } { 12 }\) \(\sin \theta = \frac { 6 } { 6.5 } = \frac { 12 } { 13 } , \cos \theta = \frac { 2.5 } { 6.5 } = \frac { 5 } { 13 } , \tan \theta = \frac { 2.5 } { 6.5 } = \frac { 5 } { 12 }\) The function values are the same because the traingles are similar and corresponding sides are proportional.
• C) \(\sin \theta = \frac { 5 } { 13 } , \cos \theta = \frac { 12 } { 13 } , \tan \theta = \frac { 5 } { 12 }\) \(\sin \theta = \frac { 2.5 } { 6.5 } = \frac { 5 } { 13 } , \cos \theta = \frac { 6 } { 6.5 } = \frac { 12 } { 13 } , \tan \theta = \frac { 2.5 } { 6 } = \frac { 5 } { 12 }\) The function values are the same because the traingles are similar and corresponding sides are proportional.
• D) \(\sin \theta = \frac { 13 } { 5 } , \cos \theta = \frac { 13 } { 12 } , \tan \theta _ { 1 } = \frac { 5 } { 12 }\) \(\sin \theta = \frac { 6.5 } { 2.5 } = \frac { 13 } { 5 } , \cos \theta = \frac { 6.5 } { 6 } = \frac { 13 } { 12 } , \tan \theta = \frac { 2.5 } { 6 } = \frac { 5 } { 12 }\) The function values are the same because the traingles are similar and corresponding sides are proportional.
• E) \(\sin \theta = \frac { 13 } { 5 } , \cos \theta = \frac { 13 } { 12 } , \tan \theta = \frac { 5 } { 12 }\) \(\sin \theta = \frac { 6.5 } { 5 } = \frac { 13 } { 2.5 } , \cos \theta = \frac { 6.5 } { 12 } = \frac { 13 } { 6 } , \tan \theta = \frac { 2.5 } { 12 } = \frac { 5 } { 6 }\) The function values are different because the traingles are not similar and corresponding sides are not proportional.