©Carolyn C. Wheater, Basis of Trigonometry uTrigonometry, or "triangle measurement," developed as a means to calculate the lengths of sides of right triangles. uIt is based upon similar triangle relationships.

©Carolyn C. Wheater, Right Triangle Trigonometry uYou can quickly prove that the two right triangles with an acute angle of 25°are similar uAll right triangles containing an angle of 25° are similar 25  You could think of this as the family of 25° right triangles. Every triangle in the family is similar. We could imagine such a family of triangles for any acute angle. You could think of this as the family of 25° right triangles. Every triangle in the family is similar. We could imagine such a family of triangles for any acute angle.

©Carolyn C. Wheater, Right Triangle Trigonometry uIn any right triangle in the family, the ratio of the side opposite the acute angle to the hypotenuse will always be the same, and the ratios of other pairs of sides will remain constant.

©Carolyn C. Wheater, The Three Main Ratios uIf the three sides of the right angle are labeled as n the hypotenuse, n the side opposite a particular acute angle, A, and n the side adjacent to the acute angle A, usix different ratios are possible. A hypotenuse adjacent opposite

©Carolyn C. Wheater, The Three Main Ratios SOH CAH TOA A c b a

©Carolyn C. Wheater, Solving Right Triangles uWith these ratios, it is possible n to solve for any unknown side of the right triangle, if another side and an acute angle are known, or n to find the angle if two sides are known. Once upon a time, students had to rely on tables to look up these values. Now the sine, cosine, and tangent of an angle can be found on your calculator.

©Carolyn C. Wheater, Trig Tables

©Carolyn C. Wheater, Sample Problem uIn right triangle ABC, hypotenuse is 6 cm long, and  A measures 32 . Find the length of the shorter leg. n Make a sketch n If one angle is 32 , the other is 58  n The shorter leg is opposite the smaller angle, so you need to find the side opposite the 32  angle  58 

©Carolyn C. Wheater, Choosing the Ratio u... Find the length of the shorter leg. n You need a ratio that talks about opposite and hypotenuse n Can use sine (sin) or cosecant (csc), but since your calculator has a key for sin, sine is more convenient  58 

©Carolyn C. Wheater, Solving the Triangle From your calculator, you can find that sin(32  )  0.53, so 6 32  58 