1. Lesson 2: TRIGONOMETRY OF RIGHT TRIANGLES Math 12 Plane and Spherical Trigonometry

2. OBJECTIVES At the end of the lesson the students are expected to: Define the six trigonometric functions as ratios of the sides of a right triangle Evaluate the trigonometric functions of an angle Evaluate the trigonometric functions of special angles Solve right triangles.

3. TRIGONOMETRIC FUNCTIONS Let 𝜃 be an acute angle in a right triangle, then 𝑠𝑖𝑛𝑒𝑠𝑖𝑛𝜃=𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑐𝑜𝑠𝑖𝑛𝑒𝑐𝑜𝑠𝜃=𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑡𝑎𝑛𝜃=𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝑐𝑜𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑐𝑜𝑡𝜃=𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑠𝑒𝑐𝑎𝑛𝑡𝑠𝑒𝑐𝜃=h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝑐𝑜𝑠𝑒𝑐𝑎𝑛𝑡𝑐𝑠𝑐𝜃=h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 hypotenuse Opposite side Adjacent side

4. RECIPROCAL FUNCTIONS The following gives the reciprocal relations of the six trigonometric functions: sin𝜃=1csc𝜃 csc𝜃=1sin𝜃 cos𝜃=1sec𝜃 sec𝜃=1cos𝜃 tan𝜃=1cot𝜃 cot𝜃=1tan𝜃

5. PYTHAGOREAN THEOREM The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Referring to the right triangle below, then 𝑐2=𝑎2+𝑏2 The Pythagorean Theorem is used to find the side of a right triangle B c a C A b

6. FUNCTIONS OF COMPLEMENTARY ANGLES B c a sin A = cos B = cos A = sin B = tan A = cot B = tan B = cot A = sec A = csc A = sec B = csc B = C A b Comparing the trigonometric functions of the acute angles A and B, and making use of the fact that A and B are complementary angles (A+B=900), then

7. FUNCTIONS OF COMPLEMENTARY ANGLES sin B = sin = cos A cos B = cos = sin A tan B = tan = cot A cot B = cot = tan A csc B = csc = sec A sec B = sec = csc A The relations may then be expressed by a single statement that: A trigonometric function of an angle is always equal to the co-function of the complement of the angle.

8. TRIGONOMETRIC FUNCTIONS OF SPECIAL ANGLES 45°, 30° and 60° To find the functions of 45°, construct an isosceles right triangle with each leg equal to 1, that is, 𝑎=1 and 𝑏=1.. By Pythagorean Theorem, the hypotenuse 𝑐=2. sin45°=12=22 cos45°=12=22 tan45°=11=1 cot45°=11=1 sec45°=21=2 csc45°=21=2 450 1 450 1

9. To find the functions of 300 and 600, take an equilateral triangle of side 2 and draw the bisector of one of the angles. This bisector divides the equilateral triangle into two congruent right triangles whose angles are 300 and 600. By Pythagorean Theorem the length of the altitude is 3. 300 2 600 1

10. sin30°=12sin60°=32 cos30°=32cos60°=12 tan30°=13=33tan60°=31=3 cot30°=31=3cot60°=13=33 sec30°=23=233sec60°=21=2 csc30°=21=2csc60°=23=233

11. EXAMPLES Draw the right triangle whose sides have the following values, and find the six trigonometric functions of the acute angle A: a) 𝑎=8, 𝑏=15 b) 𝑏=21, 𝑐=29 c) 𝑎=2, 𝑏=3 The point (5, 12) is the endpoint of the terminal side of an angle in standard position. Determine the exact value of the six trigonometric functions of the angle.

12. EXAMPLES Find the other five trigonometric functions of the acute angle A, given that: a) sin𝐴=23 b) sec𝐴=2 c) sin𝐴=2𝑚𝑛𝑚2+𝑛2 Express each of the following in terms of its cofunction: a) sin73° b) cos20°+𝐴 c) cot60°−𝛽 d) tan46°35′23"

13. EXAMPLES Determine the value of 𝛽that will satisfy the ff.: a) tan2𝛽+10°=cot3𝛽 b) sin5𝛽−10=1sec3𝛽+4° Evaluate each of the following : a) sec30°−sin60°−cos30° b) 𝑐𝑜𝑡2 45°+𝑡𝑎𝑛260°−𝑠𝑖𝑛245°