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TRIGONOMETRY MODULE: PROBLEMS 1. Fill in the table given below by the appropriate measures: Degrees 300o 150o 6 Radians 36o 3 4 7 6 2. The radius of Earth is about 6367 km. Calculate the length of the equator corresponding to the central angle of 1o 3. In the unit circle for the directed arcs in standard position given below, find the coordinates of the terminal points of each: a) 3 b) − 7 4 c) − 4 d) 41 6 e) 5 4 7 6 f) g) − 5 6 h) − 2 3 4. Find the radian measure of the angle that corresponds to the coordiantes of the terminal points on the unit circle given below: a) 3 , 1 b) 1 , 1 c) −1 , 3 d) − 3 , 1 e) −1 , 1 f) −1 ,− 1 g) 1 ,− 3 h) 0 , 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 5. In the exercises below, determine the quadrant in which the angle a) sin 0 and cos 0 b) sin 0 and cos 0 a) sin 150o cos 240 otan 210 ocot 300o 6. Evaluate 7. For the acute angle 8. Let ∈0, 2 and 4 5 . Evaluate 9. Evaluate the following trignometric ratios for 10. If tan x= 3 4 d) cos 0 and tan 0 b) sin −300 o cos−240o tan −120o cot −210o . ∈ , 2 and cos =− and x is an acute angle, then find the following: 12 : 13 a) sin x a) sin b) tan x b) tan c) cot x P T Four equal squares form the letter L as shown in figure on the left. Find tan (NTL) N L d) sec x K M 37° e) csc x A pendulum 20 cm long is moved 37° from the vertical. How much is the lower end of the pendulum raised by? G x D c) cot 12. A C c) tan 0 and sin 0 cos tan cot . 11. B satifying the inequality lies: 3 tan −sin −cos tan 2 2 2 , simplify the expression sin = E F 13. Evaluate tan 2 for ∈0 , 2 if tan = 4 3 14. Let A, B, and C be three acute angles of triangle ABC. Evaluate the following: AB C a ) sin AB−sin C b ) cos ABcos C c ) cos −sin 2 2 d ) sin 2 C AB sin 2 2 2 15. Complete the table below by indicating the appropriate sign of the following trigonometric ratios: sin cos tan cot sec 17o csc + 113o - 305o + - 262o 89o 168 + o 281o - - 352o 16. Find the maximum and minimum values of the following trigonometric expressions: a) A=3 cos x−1 c) C =1−sin 2 x b) B=2−4 sin x 17. Simplify the following trigonometric expressions: a) sin 2790 ocos 4500 ocos 7290osin 3960o 18. Evaluate sin 2 ABcos2 B A A=sin 2 19. Calculate 20. If tan B= 3 5 b) sin , where 21 19 cos sin 21 cos80 2 2 AB=90 o 3 5 sin 2 cos2 cos2 8 8 12 12 in the isosceles triangles ABC with 21. ∣AB∣=∣AC∣ , find cos A 2 22. T r T A B B A C θ r O O θ C The ray AT is tangent to the circle with the center O at the point T. If AB =1, OB =4, and ∠TCO=θ, then find tan θ = ? In the figure above, the ray AT is tangent to the circle centered at O with radius r at the point A. Moreover AT ⊥ AO . If cos θ = 3/4 , express AB in terms of r. 23. Let n∈ℤ and f : ℝ ℝ be a function so that a) Draw the graph of the function f in the interval [2,3). f x=sin x 24. You are given the graph of a) Use vertical stretching of the graph of shown below, to graph n≤xn1⇔ f x=x−n . b) State whether f is periodic or not. f x=cos x 25. You are given the graph of f x=sin x , f x=cos x a) Use horizontal stretching of the graph of x shown below, to graph g x=cos 2 g x=2sin x x b) Now use reflection of the graph of g x=cos 2 x to graph h x=−cos 2 c) Determine the amplitude, period, and phase shift of h x b) Now use horizontal translation of the graph of g x=2 sin x h x=2 sin x− 4 c) Determine the amplitude, period, and phase shift of h x to graph f(x)=cos(x) f( x) = sin( x) 2 2 1 1 -π/2 -5 -2π -3π/2 -π 0 π/2 π -5 -2π -3π/2 -π 5 3π/2 2π -π/2 0 -1 -1 -2 -2 π/2 π 5 3π/2 2π 26. Sketch the graphs of the following functions: (Use translation, reflection, compression-stretching , etc. if necessary) Also determine the amplitude, period, and phase shift (if any) of each function. 1 y=2cos x y=−cos x y=2 sin x y=sin 2 x y=sin x y=3sin 2 x− 2 4 27. Find cos x if x∈ , 3 2 and 1 x=arctan 2 28. Prove the following inequalities: sin arccos x=± 1−x 2 29. Find the following functions: sin arcsin x cosarccos x cosarcsin x=± 1−x 2 tan arctan x cot arccot x 30. Find the smallest positive value of the expressions given below if it is defined. 3 1 a) arcsin 2 1 b) arccos 2 c) arcsin h) arcsin 1 i) arcsin 3 j) arcsin − o) arcsin −1 p) arccos−1 2 d) arccos 3 2 2 k) arctan − q) arctan 0 r) arctan 1 2 e) arcsin 3 3 2 2 l) arccos0 s) arctan −1 f) arccos 2 2 m) arcsin 0 t) arctan − 3 g) arctan 3 n) arccos1 1 u) arccos− 2 31. Evaluate each of the given expressions by using right-angled triangle. 3 a) sin arccos 5 4 b) tan arcsin 5 c) sin arctan 12 5 e) cot arcsin d) cosarctan 5 7 25 f) secarccos 15 17 32. In the figure belows, find the indicated side x (Hint: Use the Law of Sines) A A 45 75 x 2 x 3 30 B C 60 B C 33. In the triangles below, find the indicated side or angle. (Hint: Use the Law of Cosines) A A 30 2 4 3 B C x B θ 6 3 B 5 A 120 x 4 C C 34. In triangle ABC, the measures a, b, and c of the sides corresponding to the angles A, B and C are given below: , if a 2 =b 2c 2− 3 bc a) Find m A c) Find m C , if c 2=a 2b2 2 ab 35. Find the area of a triangle with sides , if b 2 =a 2c 2ac b) Find m B d) Find m C , if c 2=a 2b 2 a=4, b=5, c=6 (Hint: Use Heron's formula) 36. In a triangle ABC, the measures a, b, and c of the sides correspond to the angles A, B and C . It is also given that o Find the lengths of sides b and c. (Hint: Use the Law of Tangents) m B=105 , m C =15o , and bc= 33 2 2 37. Find A=sin a – sin b cos a – cos b , if a−b= . 3 38. Evaluate sin ab , if sin acos b=1 and sin bcos a= 2. 39. 40. In the figure below, ABCD is a square with 3 DE =2 EC and m(AEB)=θ. Evaluate tan θ. A D 41. Figure below consists of three equivalent squares. Find tan θ if m(CAE)=θ. A θ B E C K G θ C 12 F θ B CB ⊥ BA , BC =12 as shown in the figure. B(0,4) C D E Find the value of cos α if m(OAC)=α. O A(3,0)