Warm Up for Section 1.4

Warm Up for Section 1.4 Find the missing edge lengths: (1). (2). 45o 60o 45o 30o Find the value of x to the nearest tenth: (3). (4). 46o x x 63o

Answers for Warm Up Section 1.4 Find the missing edge lengths: (1). (2). 45o 60o 45o 30o Find the value of x to the nearest tenth: (3). (4). 46o x x 63o

Work for Answers to WU, Section 1.4 (1). (2). (3). (4). 60o 45o 30o 45o

Trig Ratios with Complementary Angles and Similar Triangles Standard:MM2G2ab Section 1.4 Essential Question: What is the relationship between the trig ratios of complementary angles and similar triangles?

Review: With your partner answer each question reviewing the information we have learned in Sections 1.1-1.3: (1). Label each side of the triangle at right as the hypotenuse, adjacent side to ,or sideoppositeangle . (2). Using the terms “opposite,” “adjacent,” and “hypotenuse,” write each trig ratio for the triangle. (Abbreviate please!) (a). cos = ______ (b). tan  = ______ (c). sin  = ______ hypotenuse opposite adjacent Opp Adj Adj Hyp Opp Hyp

Investigation 1: Use Triangles #1, #2, and #3 below to complete the chart at the bottom of the page: In each cell of the chart you will write the indicated trig ratio for the specified angle. DO NOT simplify each fraction. Triangle #1 Triangle #2 Triangle #3

128 132 31 128 128 132 31 132 128 31 15 85 83 85 15 83 83 85 15 85 83 15 75 128 37 128 75 37 37 128 75 128 37 75

(3). What is the sum of the measures a and b in each right triangle? ____o (4). So, these angles are referred to as ______________ angles. (5). If a = 70o then b = _____o. (6). Now refer to the values in the chart. Are any of values exactly the same? ____ 90 complementary 20 Yes!

(7). Record the trig functions that have equivalent values for each triangle. The first example has been done for you for Triangle #1. Triangle #1 __________ = __________ __________ = __________ Triangle #2 and #3 __________ = __________ __________ = __________ sin acosb sin bcosa sin acosb sin bcosa Summary: For each pair of complementary angles in a right triangle, the sine of one angle is the cosine of its _____________. complement

Check for Understanding: (8). sin (30o)= cos(___) (9). cos(20o)= sin(____) (10). sin(12o) = cos(___) In general: sin  = cos (_________) cos = sin (_________) (11). If angle A is the complement of angle B and then cos B = ______. (12). If angle C is the complement of angle D and cosD = 0.887, then sin C = ________. 70o 60o 78o 90o –  90o –  sin A 0.887

Investigation 2: The multiplicative inverse, or reciprocal, of any real number a where a 0 is . So, the reciprocal of 2 is and the reciprocal of is _____. Look at the chart again and the values for the tangent ratio in each triangle. Fill in the blank for each triangle. The first blank has been done for you.

Triangle #1 Triangle #2 and #3 __________ = __________ __________ = __________ __________ = __________ __________ = __________ tan a tan a tan b tan b Summary:For each pair of complementary angles in a right triangle, the tangent of one angle is the ________ of the tangent of its complement. reciprocal

Check for Understanding: (13). tan (20o) = ________ (14). tan (40o) = ______ (15). tan (32o) = _________ In general, tan  = _____________. (16). If A and B are complementary angles and tan A = , then tan B = _______. (17). If C and D are complementary angles and tan C = 2.2, then tan D = .

Write an identity statement using the complement of the given angle. (18). sin 52o = (19). cos 29o = (20). tan 60o = (21). cos 75o = (22). tan 10o = (23). sin 67o = cos 38o sin 61o sin 15o cos 23o

Y 15 5 9 4 A Z X 12 3 B C Investigation 3: (24). Do ∆ABC and ∆XYZ appear to be similar triangles? _____ YES! (25). Let’s check the ratio of the corresponding sides:

Y 15 5 9 4 A Z X 12 3 B C Investigation 3: (26). Does this confirm that the triangles are similar? ______ YES!

Y A 15 9 5 4 Z X 12 3 B C (27). sin A = sin X = cos A = cos X = tan A = tan X = (28). Are the sine, cosine, and tangent ratios for corresponding angles of similar triangles the same? YES!

39 36 R 15 T S Check for Understanding: Given:∆RST  ∆PYQ Q 52 20 Y P 48 Fill in each blank with the corresponding angle: (29). sin R = sin ___ (30). cos Q = cos ___ (31). tan T = tan ___ P T Q

39 36 R 15 T S Check for Understanding: Given:∆RST  ∆PYQ Q 52 20 Y P 48 Write each ratio as a fraction in simplest form: (32). sin R = _______ and sin P = _______ (33). tan T = _______ and tan Q = _______

(34). If DOG CAT with right angles at O and at A , then sin D = ______, tan T = ______, and cos G = ______. Classwork Answers: Write each trig expression as an equivalent expression involving the complement of the angle. (35). sin(45o) = ________ (36). cos(50o) = _______ (37). tan(22o) = ________ (38). cos(70o) = _______ (39). tan(31o) = _________ (40). sin(49o) = _______ sin C tan G cos T cos (45o) sin (40o) 1 sin (20o) tan (68o) 1 cos (41o) tan (59o)

Angle A is the complement of angle B. Find each of the following. (41). (42). (43). (44). (45).(46). (47).

Formula Sheet: Complementary Angles: sin θ = _______________ cosθ = ________________ tan θ = ________________ sin(90° –θ) cos(90° –θ) tan(90° –θ)

X A B C Y Z Similar Triangles: (ΔABC ~ ΔXYZ) sin A = _______ cosA = _______ tan A =_______ sin C= _______ cos C = _______ tan C =_______ sin X cos X tan X sin Z cos Z tan Z

Warm Up for Section 1.4