Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios
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1 Eercise Set.1: Special Right Triangles and Trigonometric Ratios Answer the following If two sides of a triangle are congruent, then the opposite those sides are also congruent. 2. If two angles of a triangle are congruent, then the opposite those angles are also congruent. 3. In an triangle, the sum of the measures of its angles is degrees In an isosceles right triangle, each acute angle measures degrees.. Fill in each missing blank with one of the smallest, largest In an triangle, the longest side is opposite the angle, and the shortest side is opposite the angle. 6. Fill in each missing blank with one of the,, 90 o In a o triangle, the hpotenuse is opposite the angle, the shorter leg is opposite the angle, and the longer leg is opposite the angle For each of the following, (a) Use the theorem for o triangles to find. (b) Use the Pthagorean Theorem to verif the result obtained in part (a) The Universit of Houston
2 Eercise Set.1: Special Right Triangles and Trigonometric Ratios In the figure below, an altitude is drawn to the base of an equilateral triangle. (a) Find a and b. (b) Justif the answer obtained in part (a). (c) Use the Pthagorean Theorem to find c, the length of the altitude. (Write c in simplest radical form.) a c b 1. For each of the following, Use the theorem for o triangles to find and The following eamples help to illustrate the theorem regarding o triangles. 19. What is the measure of each angle of an equilateral triangle? 20. An altitude is drawn to the base of the equilateral triangle drawn below. Find the measures of and o 2. o In the figure below, an altitude is drawn to the base of an equilateral triangle. (a) Find a and b. (b) Justif the answer obtained in part (a). (c) Use the Pthagorean Theorem to find c, the length of the altitude. 26. c 10 a b The Universit of Houston
3 Eercise Set.1: Special Right Triangles and Trigonometric Ratios 2. (b) Find the sin cos tan ( A ) sin ( ) ( A ) cos ( ) ( A ) tan ( ) B B B D E (a) Use the Pthagorean Theorem to find DE. (b) Find the sin cos tan 2 F ( D ) sin ( ) ( D ) cos ( ) ( D ) tan ( ) F F F Suppose that θ is an acute angle of a right triangle and sin ( θ ) =. Find cos( θ ) and tan ( θ ). 36. Suppose that θ is an acute angle of a right 2 triangle and tan ( θ ) =. Find sin ( θ ) and cos( θ ) The reciprocal of the sine function is the 3. The reciprocal of the cosine function is the Answer the following. Write answers in simplest form. 39. The reciprocal of the tangent function is the 33. A The reciprocal of the cosecant function is the 1. The reciprocal of the secant function is the C (a) Use the Pthagorean Theorem to find BC. The Universit of Houston B 2. The reciprocal of the cotangent function is the
4 Eercise Set.1: Special Right Triangles and Trigonometric Ratios 3. 6 β (a) Use the Pthagorean Theorem to find. (b) Find the si trigonometric functions of. (c) Find the si trigonometric functions of β.. Suppose that θ is an acute angle of a right 2 10 triangle and cot ( θ ) =. Find the si 3 trigonometric functions of θ.. Suppose that θ is an acute angle of a right triangle and sec( θ ) =. Find the si 2 trigonometric functions of θ β (a) Use the Pthagorean Theorem to find. (b) Find the si trigonometric functions of. (c) Find the si trigonometric functions of β. β (a) Use the Pthagorean Theorem to find. (b) Find the si trigonometric functions of. (c) Find the si trigonometric functions of β. β 6 (a) Use the Pthagorean Theorem to find. (b) Find the si trigonometric functions of. (c) Find the si trigonometric functions of β (a) Use the theorems for special right triangles to find the missing side lengths in the triangles above. (b) Using the triangles above, find the csc ) sec ) cot ) sin cos tan (c) Using the triangles above, find the csc 30 ) sec 30 ) cot 30 ) sin 30 cos 30 tan 30 (d) Using the triangles above, find the csc 60 ) sec 60 ) cot 60 ) sin 60 cos 60 tan 60 2 The Universit of Houston
5 Eercise Set.1: Special Right Triangles and Trigonometric Ratios (a) Use the theorems for special right triangles to find the missing side lengths in the triangles above. (b) Using the triangles above, find the csc ) sec ) cot ) sin cos tan (c) Using the triangles above, find the csc 30 ) sec 30 ) cot 30 ) sin 30 cos 30 tan 30 (d) Using the triangles above, find the csc 60 ) sec 60 ) cot 60 ) sin 60 cos 60 tan Compare the answers to parts (b), (c), and (d) in the previous two eamples. What do ou notice? The Universit of Houston
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