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Right Angle Trigonometry I. Basic Facts and Definitions 1. Right angle – angle measuring 90 2. Straight angle – angle measuring 180 3. Acute angle – angle measuring between 0 and 90 4. Complementary angles – two angles whose sum is 90 5. Supplementary angles – two angles whose sum is 180 6. Right triangle – triangle with a right angle 7. Isosceles triangle – a triangle with exactly two sides equal 8. Equilateral triangle – a triangle with all three sides equal 9. The sum of the angles of a triangle is 180 . 10. In general, capital letters refer to angles while small letters refer to the sides of a triangle. For example, side a is opposite angle A . II. Right Triangle Facts and Examples 1. Hypotenuse – the side opposite the right angle, side c . 2. Pythagorean Theorem - a 2 b 2 c 2 3. A and B are complementary. www.rit.edu/asc Page 1 of 12 III. Examples: 1. In a right triangle, the hypotenuse is 10 inches and one side is 8 inches. What is the length of the other side? Solution: B a 2 b2 c 2 82 b 2 10 2 64 b 2 100 a 8 b 36 2 c 10 b6 C b? A 2. In a right triangle ABC , if A 23 , what is the measure of B ? Solution: The two acute angles in a right triangle are complementary. A B 90 23 B 90 B 67 IV. B C Similar Triangles: a. Two triangles are similar if the angles of one triangle are equal to the corresponding angles of the other. In similar triangles, ratios of corresponding sides are equal. A Conditions for Similar Triangles ( ABC ~ EGF ) 1. Corresponding angles in similar triangles are equal: A E B F C G 2. Ratios of corresponding sides are equal: AC AB BC EG EF FG www.rit.edu/asc Page 2 of 12 Example 1: AE 50 meters EF 22 m and AB 100 m Find the length of side BC . A Notice that ABC and AEF are similar since corresponding angles are equal. (There is a right angle at both F and C , A is the same in both triangles and B equals the acute angle at E .) 50 F E 22 50 C B Solution: AE EF 50 22 so AB BC 100 BC By cross multiplying we get: 50( BC ) 22(100) Therefore BC 44 meters. AE 50 meters EF 22 m and AB 100 m Find the length of side BC . Example 2: All 45 45 90 triangles are similar to one another. Two sides are of equal length and the hypotenuse is 2 times the length of each of the equal sides. All 30 60 90 triangles are similar to one another. The shortest side of length a is opposite the smallest angle ( 30 ). The hypotenuse is twice the length of the shortest side. The side opposite the 60 has a length 3 times the shorter leg. www.rit.edu/asc Page 3 of 12 Problem: Find the lengths of the legs of a 30 60 90 triangle if the hypotenuse is 8 meters. Solution: 1) If 2a 8 , then a 4 meters and 2) V. 3a 3(4) 4 3 meters. The Six Trigonometric Ratios for Acute Angles B c a C A b opposite a hypotenuse c adjacent b cosine A cos A hypotenuse c opposite a tangent A tan A adjacent b 1 hypotenuse c sin A opposite a 1 hypotenuse c secant A sec A cos A adjacent b 1 adjacent b cotangent A cot A tan A opposite a sine A sin A cosecant A csc A TRIG TRICK: A good way to remember the trig ratios is to use the mnemonic SOH CAH TOA! SOH CAH TOA i p n p e o s i t e y p o t e n u s e o s i n e d j a c e n t y p o t e n u s e www.rit.edu/asc a n g e n t p p o s i t e d j a c e n t Page 4 of 12 Example 1: Find the six trigonometric ratios for the acute angle B . Solution: opposite b . Using the above definitions, the rest are: hypotenuse c a b c c a cos B , tan B , csc B , sec B , cot B c a b a b sin B Example 2: In the right ABC , a 1 and b 3 . Determine the six trigonometric ratios for B . Solution: Use Pythagorean Theorem: c2 a2 b2 c 2 12 32 c 2 10 2 c 10 (Since length is positive, we will only use c 10 .) sin B opp 3 3 10 hyp 10 10 cos B adj 1 10 hyp 10 10 tan B opp 3 3 adj 1 csc B hyp 10 opp 3 sec B hyp 10 10 adj 1 cot B adj 1 opp 3 www.rit.edu/asc Page 5 of 12 VI. Special Cases a. Trigonometric values of 30 and 60 (Use the 30 60 90 triangle from pg. 3.) sin 30 30 c2 a 3 60 60 b 1 1 2 3 2 1 3 tan 30 3 3 cos 30 csc 30 3 2 1 cos 60 2 3 tan 60 3 1 sin 60 2 2 1 csc 60 2 3 sec 30 2 2 3 3 3 sec 60 cot 30 3 3 1 cot 60 2 3 3 2 2 1 1 3 3 3 b. Trigonometric values of 45 (Use the 45 45 90 triangle from pg. 3.) a 1 45 c 2 45 b 1 sin 45 1 2 2 2 csc 45 cos 45 1 2 2 2 sec 45 1 tan 45 1 1 www.rit.edu/asc 2 2 1 2 2 1 1 cot 45 1 1 Page 6 of 12 VII. Converting Minutes and Seconds to Decimal Form (Necessary for most calculator use in evaluating trig values) 1. 2. To convert from seconds to a decimal part of a minute, divide the number of seconds by 60. To convert from minutes to a decimal part of a degree, divide the number of minutes by 60. Example 1: Convert 6447 to degrees using decimals. Solution: 47 6447 64 60 6447 64 .783 64.783 Example 2: Convert 151210 to degrees using decimals. Solution: 151210 1512 10 10 151210 1512 60 151210 1512.167 12.167 151210 15 60 151210 15 .203 15.203 VIII. Right Triangle Trigonometry Problems To Solve Right Triangle Problems: There are six parts to any triangle; 3 sides and 3 angles. Each trig formula (ex: sin A = a/c) contains three parts; one acute angle and two sides. If you know values for two of the three parts then you can solve for the third unknown part using the following method: 1. Draw a right triangle. Label the known parts with the given values and indicate the unknown part(s) with letters. 2. To find an unknown part, choose a trig formula which involves the unknown part and the two known parts. www.rit.edu/asc Page 7 of 12 Example 1: A right triangle has a 38 and B 61 . Find the length of side b . Solution: Which trig formulas involve an acute angle (B) and the side opposite (b) and the side adjacent (a) to the angle? Since both tangent and cotangent do, either could be used to solve this problem. We will use tangent. A b C opp b b b so, tan 61 or 1.8040 . adj a 38 38 Therefore b 68.6 . tan B 61 a 38 B (Always check your answer by comparing size of angle and length of side; the longer side is always opposite the larger angle.) IX. Angles of Elevation and Depression Angle of depression Angle of elevation Example: From a point 124 feet from the foot of a tower and on the same level, the angle of elevation of the tower is 3620 . Find the height of the tower. Solution: h tan 3620 tan(36.333) 24 h 124(0.7355) h 3620 124 ft. h 91.2 ft. www.rit.edu/asc Page 8 of 12 Practice Problems: 1. In right triangle ABC , if c 39 inches and b 36 inches, find a . 2. Find the length of side AC . Note: This problem and diagram corresponds to finding the height of a street light pole ( AC ) if a 6 ft. man ( EF ) casts a shadow ( BF ) of 15 ft. and the pole casts a shadow ( BC ) of 45 ft. A E 6 C 30 15 B F 3. Evaluate: F 13 12 G 5 E a) sin E = _____________ b) tan E = _____________ c) cos F = _____________ d) sec F = _____________ 4. Evaluate: (Draw reference 30 60 90 and 45 45 90 triangles) a) sin 30 = _____________ b) tan 60 = _____________ c) sec 60 = _____________ d) tan 45 = _____________ e) csc 45 = _____________ f) cot 30 = _____________ www.rit.edu/asc Page 9 of 12 5. Evaluate: B 10 8 C A a) tan A = _____________ b) csc B = _____________ c) cot A = _____________ d) sec B = _____________ 6. Convert to decimal notation using a calculator: a) 7606 b) 452713 Evaluate, using a calculator: c) sin 5248 d) cot 3942 7. Label the sides and remaining angles of right triangle ABC , using A , B , a , b and c . If a 43 and A 37 , find the values of the remaining parts. C www.rit.edu/asc Page 10 of 12 8. Given right triangle ABC with b 0.622 and A 5140 , find c . Draw a diagram. 9. From a cliff 140 feet above the shore line, an observer notes that the angle of depression of a ship is 2130 . Find the distance from the ship to a point on the shore directly below the observer. cliff ship www.rit.edu/asc Page 11 of 12 Answers to Right Triangle Trigonometry: 1. a 15 inches (use Pythagorean Theorem) 2. AC BC EF BF 3. a) sin E AC 6 45 15 12 13 b) tan E AC 18 12 5 c) cos F 4. (see part E of the handout)= a) sin 30 1 b) tan 60 3 2 d) tan 45 1 e) csc 45 2 12 13 7. B 53 b) 45.454 b 57.06 13 12 c) sec 60 2 f) cot 30 3 5. b 6 (use Pythagorean Theorem) 8 4 5 3 a) tan A b) csc B c) cot A 6 3 3 4 6. a) 76.1 d) sec F c) .7965 d) sec B 5 4 d) 1.2045 c 71.45 B c a C b 8. c 1 (Use cos A A b c or sec A to solve for unknown c ) c b 140 x x 355.41 ft. 9. tan 2130 (Angle of depression) cliff ship www.rit.edu/asc Page 12 of 12