Section 1.4
Trigonometric Functions and their Inverses Key rules Degrees to radians: multiply by π 180 \frac{\pi}{180} 180 π . Radians to degrees: multiply by 180 π \frac{180}{\pi} π 180 . Arc length s = r θ s = r\theta s = r θ ( θ \theta θ in radians; clockwise angles are negative). Reference angle gives the value, and the quadrant gives the sign: Q I all positive, Q II only sin , csc \sin, \csc sin , csc , Q III only tan , cot \tan, \cot tan , cot , Q IV only cos , sec \cos, \sec cos , sec . If P ( x , y ) P(x, y) P ( x , y ) is on the terminal side, r = x 2 + y 2 r = \sqrt{x^2 + y^2} r = x 2 + y 2 , sin θ = y r \sin\theta = \frac{y}{r} sin θ = r y , cos θ = x r \cos\theta = \frac{x}{r} cos θ = r x , tan θ = y x \tan\theta = \frac{y}{x} tan θ = x y . Identities: sin 2 θ + cos 2 θ = 1 \sin^2\theta + \cos^2\theta = 1 sin 2 θ + cos 2 θ = 1 , 1 + tan 2 θ = sec 2 θ 1 + \tan^2\theta = \sec^2\theta 1 + tan 2 θ = sec 2 θ , 1 + cot 2 θ = csc 2 θ 1 + \cot^2\theta = \csc^2\theta 1 + cot 2 θ = csc 2 θ ; cofunctions sin θ = cos ( 90 ∘ − θ ) \sin\theta = \cos(90^\circ - \theta) sin θ = cos ( 9 0 ∘ − θ ) ; even/odd: cos ( − θ ) = cos θ \cos(-\theta) = \cos\theta cos ( − θ ) = cos θ , sin ( − θ ) = − sin θ \sin(-\theta) = -\sin\theta sin ( − θ ) = − sin θ , tan ( − θ ) = − tan θ \tan(-\theta) = -\tan\theta tan ( − θ ) = − tan θ . Sum and double angle: sin ( A ± B ) = sin A cos B ± cos A sin B \sin(A \pm B) = \sin A\cos B \pm \cos A\sin B sin ( A ± B ) = sin A cos B ± cos A sin B , cos ( A ± B ) = cos A cos B ∓ sin A sin B \cos(A \pm B) = \cos A\cos B \mp \sin A\sin B cos ( A ± B ) = cos A cos B ∓ sin A sin B , sin 2 θ = 2 sin θ cos θ \sin 2\theta = 2\sin\theta\cos\theta sin 2 θ = 2 sin θ cos θ , cos 2 θ = 2 cos 2 θ − 1 \cos 2\theta = 2\cos^2\theta - 1 cos 2 θ = 2 cos 2 θ − 1 . Ranges: sin − 1 x ∈ [ − π 2 , π 2 ] \sin^{-1}x \in [-\frac{\pi}{2}, \frac{\pi}{2}] sin − 1 x ∈ [ − 2 π , 2 π ] , cos − 1 x ∈ [ 0 , π ] \cos^{-1}x \in [0, \pi] cos − 1 x ∈ [ 0 , π ] , tan − 1 x ∈ ( − π 2 , π 2 ) \tan^{-1}x \in (-\frac{\pi}{2}, \frac{\pi}{2}) tan − 1 x ∈ ( − 2 π , 2 π ) . So cos − 1 ( cos θ ) = θ \cos^{-1}(\cos\theta) = \theta cos − 1 ( cos θ ) = θ only when θ ∈ [ 0 , π ] \theta \in [0, \pi] θ ∈ [ 0 , π ] . Visual guide Unit circle with the points for 0, π/6, π/4, π/3, π/2, π and 3π/2: (1, 0), (√3/2, 1/2), (√2/2, √2/2), (1/2, √3/2), (0, 1), (−1, 0) and (0, −1). Quadrant signs: Q I all positive, Q II sin, Q III tan, Q IV cos. A Q I: all positive S Q II: sin, csc positive T Q III: tan, cot positive C Q IV: cos, sec positive (1, 0) 0 (√3/2, 1/2) π/6 (√2/2, √2/2) π/4 (1/2, √3/2) π/3 (0, 1) π/2 (−1, 0) π (0, −1) 3π/2 The unit circle. Each point is (cos θ, sin θ). The letters A, S, T, C show which functions are positive in each quadrant. One period of y = sin x from 0 to 2π, with key points (0, 0), (π/2, 1), (π, 0), (3π/2, −1) and (2π, 0). y = sin x π/2 π 3π/2 2π 1 −1 (π/2, 1) (3π/2, −1) One period of y = cos x from 0 to 2π, with key points (0, 1), (π/2, 0), (π, −1), (3π/2, 0) and (2π, 1). y = cos x π/2 π 3π/2 2π 1 −1 (0, 1) (π, −1) (2π, 1) One period of y = sin x and y = cos x. The key points match the exact values on the unit circle, for example sin(π/2) = 1 and cos(π) = −1.
Questions 94 total 1 Book example Example 1 (a) Convert the degree measure to radians: 75 ∘ 75^\circ 7 5 ∘ 2 Book example Example 1 (b) Convert the degree measure to radians: − 210 ∘ -210^\circ − 21 0 ∘ 3 Book example Example 2 (a) Convert the radian measure to degrees: 7 π 3 \frac{7\pi}{3} 3 7 π 4 Book example Example 2 (b) Convert the radian measure to degrees: − 11 π 4 -\frac{11\pi}{4} − 4 11 π 5 Book example Example 3 In the right triangle in the figure, the side opposite θ \theta θ has length 3 3 3 and the hypotenuse has length 5 5 5 . Find all six trigonometric functions of θ \theta θ and of the other acute angle α \alpha α . 6 Book example Example 4, part 1 Find the reference angle of 150 ∘ 150^\circ 15 0 ∘ . 7 Book example Example 4, part 2 Find the reference angle of 225 ∘ 225^\circ 22 5 ∘ . 8 Book example Example 4, part 3 Find the reference angle of 11 π 6 \frac{11\pi}{6} 6 11 π . 9 Book example Example 4, part 4 Find the reference angle of 11 π 3 \frac{11\pi}{3} 3 11 π . 10 Book example Example 5, part 3 Find the exact values of all six trigonometric functions of θ = 225 ∘ \theta = 225^\circ θ = 22 5 ∘ . 11 Book example Example 5, part 5 Find the exact values of all six trigonometric functions of θ = 17 π 6 \theta = \frac{17\pi}{6} θ = 6 17 π . 12 Related problem Related Problem 4, part 1 Find the values of all six trigonometric functions of θ = π 4 \theta = \frac{\pi}{4} θ = 4 π . 13 Related problem Related Problem 4, part 2 Find the values of all six trigonometric functions of θ = 2 π 3 \theta = \frac{2\pi}{3} θ = 3 2 π . 14 Related problem Related Problem 4, part 3 Find the values of all six trigonometric functions of θ = − 7 π 4 \theta = -\frac{7\pi}{4} θ = − 4 7 π . 15 Book example Example 6, part 1 If θ \theta θ is in standard position and the point P ( 3 , 4 ) P(3, 4) P ( 3 , 4 ) is on the terminal side of θ \theta θ , find the values of all six trigonometric functions of θ \theta θ . 16 Book example Required Example 6, part 2 If θ \theta θ is in standard position and the point P ( − 3 , 4 ) P(-3, 4) P ( − 3 , 4 ) is on the terminal side of θ \theta θ , find the values of all six trigonometric functions of θ \theta θ . 17 Book example Example 6, part 3 If θ \theta θ is in standard position and the point P ( 2 , − 12 ) P(2, -\sqrt{12}) P ( 2 , − 12 ) is on the terminal side of θ \theta θ , find the values of all six trigonometric functions of θ \theta θ . 18 Book example Example 7 Using the Pythagorean identity, if sin θ = 3 5 \sin\theta = \frac{3}{5} sin θ = 5 3 and θ \theta θ lies in Quadrant I, find the values of cos θ \cos\theta cos θ and tan θ \tan\theta tan θ . 19 Book example Example 8 Use the Pythagorean identities to verify the identity 1 − sin 2 θ 1 + tan 2 θ = cos 4 θ \frac{1 - \sin^2\theta}{1 + \tan^2\theta} = \cos^4\theta 1 + t a n 2 θ 1 − s i n 2 θ = cos 4 θ . 20 Book example Example 9, part 2 Use the cofunction identities to verify sin ( 90 ∘ − θ ) − cos θ sin θ + cos ( 90 ∘ − θ ) = 0 \frac{\sin(90^\circ - \theta) - \cos\theta}{\sin\theta + \cos(90^\circ - \theta)} = 0 s i n θ + c o s ( 9 0 ∘ − θ ) s i n ( 9 0 ∘ − θ ) − c o s θ = 0 . 21 Book example Required Example 10, part 1 Use the identities for even and odd trigonometric functions to verify sin ( − θ ) + cos ( − θ ) = cos θ − sin θ \sin(-\theta) + \cos(-\theta) = \cos\theta - \sin\theta sin ( − θ ) + cos ( − θ ) = cos θ − sin θ . 22 Book example Example 10, part 2 Use the identities for even and odd trigonometric functions to verify sin ( − θ ) tan ( − θ ) cos ( − θ ) = tan 2 θ \frac{\sin(-\theta)\tan(-\theta)}{\cos(-\theta)} = \tan^2\theta c o s ( − θ ) s i n ( − θ ) t a n ( − θ ) = tan 2 θ . 23 Book example Example 10, part 3 Use the identities for even and odd trigonometric functions to verify csc ( − θ ) cot ( − θ ) + sec ( − θ ) cos ( − θ ) = csc θ cot θ + 1 \csc(-\theta)\cot(-\theta) + \sec(-\theta)\cos(-\theta) = \csc\theta\cot\theta + 1 csc ( − θ ) cot ( − θ ) + sec ( − θ ) cos ( − θ ) = csc θ cot θ + 1 . 24 Related problem Related Problem 6, part 1 Find the value of sin − 1 ( − 1 2 ) \sin^{-1}\left(-\frac{1}{2}\right) sin − 1 ( − 2 1 ) . 25 Related problem Related Problem 6, part 2 Find the value of cos − 1 ( − 2 2 ) \cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) cos − 1 ( − 2 2 ) . 26 Related problem Related Problem 6, part 3 Find the value of cos − 1 ( sin ( − π 3 ) ) \cos^{-1}\left(\sin\left(-\frac{\pi}{3}\right)\right) cos − 1 ( sin ( − 3 π ) ) . 27 Related problem Related Problem 6, part 4 Find the value of sin − 1 ( cos ( 7 π 6 ) ) \sin^{-1}\left(\cos\left(\frac{7\pi}{6}\right)\right) sin − 1 ( cos ( 6 7 π ) ) . 28 Book exercise Required Exercises 1.4, #25 Verify the identity cos x 1 − sin x + cos x 1 + sin x = 2 sec x \frac{\cos x}{1 - \sin x} + \frac{\cos x}{1 + \sin x} = 2\sec x 1 − s i n x c o s x + 1 + s i n x c o s x = 2 sec x . 29 Book exercise Required Challenging Problems #4 Show that, for x ≥ 0 x \ge 0 x ≥ 0 , sin ( 2 cos − 1 ( x 1 + x 2 ) ) = 2 x 1 + x 2 \sin\left(2\cos^{-1}\left(\frac{x}{\sqrt{1 + x^2}}\right)\right) = \frac{2x}{1 + x^2} sin ( 2 cos − 1 ( 1 + x 2 x ) ) = 1 + x 2 2 x . 30 Book exercise Applications #2 In Riyadh, a surveillance camera is mounted on a tower 20 20 20 meters tall. A car is moving away from the base of the tower. At a certain moment, θ = tan − 1 ( 4 3 ) \theta = \tan^{-1}\left(\frac{4}{3}\right) θ = tan − 1 ( 3 4 ) , where θ \theta θ is the angle at the car between the ground and the line to the camera. Find the horizontal distance of the car from the base of the tower and the distance from the car to the top of the tower. 31 Tutorial Tutorial 4, Q1 1 Find the exact value without using a calculator: cos 225 ∘ \cos 225^\circ cos 22 5 ∘ 32 Tutorial Tutorial 4, Q1 2 Find the exact value without using a calculator: sin ( 17 π 3 ) \sin\left(\frac{17\pi}{3}\right) sin ( 3 17 π ) 33 Tutorial Tutorial 4, Q1 3 Find the exact value without using a calculator: cos 17.5 ∘ ⋅ cos 42.5 ∘ − sin 17.5 ∘ ⋅ sin 42.5 ∘ \cos 17.5^\circ \cdot \cos 42.5^\circ - \sin 17.5^\circ \cdot \sin 42.5^\circ cos 17. 5 ∘ ⋅ cos 42. 5 ∘ − sin 17. 5 ∘ ⋅ sin 42. 5 ∘ 34 Tutorial Tutorial 4, Q1 4 Find the exact value without using a calculator: tan 22.5 ∘ \tan 22.5^\circ tan 22. 5 ∘ 35 Tutorial Tutorial 4, Q2 1 Find the values of the remaining trigonometric functions: tan x = − 8 5 \tan x = -\frac{8}{5} tan x = − 5 8 and sin x < 0 \sin x < 0 sin x < 0 . 36 Tutorial Tutorial 4, Q2 2 Find the values of the remaining trigonometric functions: sin θ = 3 7 \sin\theta = \frac{3}{7} sin θ = 7 3 , where π 2 < θ < π \frac{\pi}{2} < \theta < \pi 2 π < θ < π . 37 Tutorial Tutorial 4, Q3 A Verify the identity sin x 1 − cos x = csc x + cot x \frac{\sin x}{1 - \cos x} = \csc x + \cot x 1 − c o s x s i n x = csc x + cot x . 38 Tutorial Tutorial 4, Q3 B1 Determine the exact value: sin − 1 ( − 3 2 ) \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) sin − 1 ( − 2 3 ) 39 Tutorial Tutorial 4, Q3 B2 Determine the exact value: sin ( tan − 1 ( 2 ) + tan − 1 ( 3 ) ) \sin\left(\tan^{-1}(2) + \tan^{-1}(3)\right) sin ( tan − 1 ( 2 ) + tan − 1 ( 3 ) ) 40 Past midterm Midterm 1444H S1, Q1 A If cos x = 4 9 \cos x = \frac{4}{9} cos x = 9 4 and tan x > 0 \tan x > 0 tan x > 0 , find the value of csc x \csc x csc x . 41 Past midterm Midterm 1444H S1, Q1 B(i) Find the exact value without using a calculator: cos − 1 ( cos 11 π 6 ) \cos^{-1}\left(\cos\frac{11\pi}{6}\right) cos − 1 ( cos 6 11 π ) 42 Past midterm Midterm 1444H S1, Q1 B(ii) Find the exact value without using a calculator: sin 70 ∘ cos 10 ∘ − cos 70 ∘ sin 10 ∘ \sin 70^\circ \cos 10^\circ - \cos 70^\circ \sin 10^\circ sin 7 0 ∘ cos 1 0 ∘ − cos 7 0 ∘ sin 1 0 ∘ 43 Past midterm Midterm 1444H S1, Q1 C What is the radian measure of an angle θ \theta θ opposite an arc length 24 24 24 m, in a circle of diameter 12 12 12 m, where θ \theta θ is in standard position and clockwise? 44 Past midterm Alternative Midterm 1444H S1, Q1 A If 3 cos x + 2 = 0 3\cos x + 2 = 0 3 cos x + 2 = 0 and π 2 < x < π \frac{\pi}{2} < x < \pi 2 π < x < π , find the value of cos ( 2 x ) \cos(2x) cos ( 2 x ) . 45 Past midterm Alternative Midterm 1444H S1, Q1 B(i) Find the exact value without using a calculator: sin ( cos − 1 ( 1 3 ) + tan − 1 ( 3 ) ) \sin\left(\cos^{-1}\left(\frac{1}{3}\right) + \tan^{-1}(3)\right) sin ( cos − 1 ( 3 1 ) + tan − 1 ( 3 ) ) 46 Past midterm Alternative Midterm 1444H S1, Q1 B(ii) Find the exact value without using a calculator: sin 75 ∘ \sin 75^\circ sin 7 5 ∘ 47 Past midterm Alternative Midterm 1444H S1, Q1 C If the measure in degrees of θ \theta θ is 120 ∘ 120^\circ 12 0 ∘ , where θ \theta θ is in standard position, and the diameter is 12 12 12 cm, what is the arc length opposite to θ \theta θ ? 48 Past midterm Midterm 1444H S2, Q1 A Prove the identity cos ( π 2 − x ) 1 − cos 2 x = csc x \frac{\cos\left(\frac{\pi}{2} - x\right)}{1 - \cos^2 x} = \csc x 1 − c o s 2 x c o s ( 2 π − x ) = csc x . 49 Past midterm Midterm 1444H S2, Q1 B(i) Find the exact value without using a calculator: sin − 1 ( sin 7 π 6 ) \sin^{-1}\left(\sin\frac{7\pi}{6}\right) sin − 1 ( sin 6 7 π ) 50 Past midterm Midterm 1444H S2, Q1 B(ii) Find the exact value without using a calculator: cos ( tan − 1 ( 2 3 ) + sin − 1 ( 4 5 ) ) \cos\left(\tan^{-1}\left(\frac{2}{3}\right) + \sin^{-1}\left(\frac{4}{5}\right)\right) cos ( tan − 1 ( 3 2 ) + sin − 1 ( 5 4 ) ) 51 Past midterm Midterm 1444H S2, Q1 C If θ \theta θ is in standard position and Q ( − 3 , − 6 ) Q(-3, -6) Q ( − 3 , − 6 ) is on the terminal side of θ \theta θ , find sin ( 2 θ ) \sin(2\theta) sin ( 2 θ ) . 52 Past midterm Midterm 1446H S1, Q3 A Verify the identity tan 2 x ⋅ sin 2 x = tan 2 x − sin 2 x \tan^2 x \cdot \sin^2 x = \tan^2 x - \sin^2 x tan 2 x ⋅ sin 2 x = tan 2 x − sin 2 x . 53 Past midterm Midterm 1446H S1, Q3 B(i) Find the exact value without using a calculator: cos ( − 4 π 3 ) \cos\left(-\frac{4\pi}{3}\right) cos ( − 3 4 π ) 54 Past midterm Midterm 1446H S1, Q3 B(ii) Find the exact value without using a calculator: tan ( cos − 1 ( − 3 5 ) + tan − 1 ( − 4 7 ) ) \tan\left(\cos^{-1}\left(-\frac{3}{5}\right) + \tan^{-1}\left(-\frac{4}{7}\right)\right) tan ( cos − 1 ( − 5 3 ) + tan − 1 ( − 7 4 ) ) 55 Past midterm Alternative Midterm 1446H S1, Q3 A Verify the identity 2 cos 2 x − 1 cos x − sin x = cos x + sin x \frac{2\cos^2 x - 1}{\cos x - \sin x} = \cos x + \sin x c o s x − s i n x 2 c o s 2 x − 1 = cos x + sin x . 56 Past midterm Alternative Midterm 1446H S1, Q3 B(i) Find the exact value without using a calculator: cos − 1 ( ( sec ( 2 3 ) ) − 1 ) \cos^{-1}\left(\left(\sec\left(\frac{2}{3}\right)\right)^{-1}\right) cos − 1 ( ( sec ( 3 2 ) ) − 1 ) 57 Past midterm Alternative Midterm 1446H S1, Q3 B(ii) Find the exact value without using a calculator: sin ( 2 sin − 1 ( − 4 9 ) ) \sin\left(2\sin^{-1}\left(-\frac{4}{9}\right)\right) sin ( 2 sin − 1 ( − 9 4 ) ) 58 Past midterm Midterm 1446H (photo), Q3 A Verify the identity tan θ + cos θ 1 + sin θ = sec θ \tan\theta + \frac{\cos\theta}{1 + \sin\theta} = \sec\theta tan θ + 1 + s i n θ c o s θ = sec θ . 59 Past midterm Midterm 1446H (photo), Q3 B(i) Find the exact value without using a calculator: cos ( 4 tan − 1 ( − 1 ) ) \cos\left(4\tan^{-1}(-1)\right) cos ( 4 tan − 1 ( − 1 ) ) 60 Past midterm Midterm 1446H (photo), Q3 B(ii) Find the exact value without using a calculator: sin ( sin − 1 ( 2 3 ) + 2 sin − 1 ( − 1 7 ) ) \sin\left(\sin^{-1}\left(\frac{2}{3}\right) + 2\sin^{-1}\left(-\frac{1}{7}\right)\right) sin ( sin − 1 ( 3 2 ) + 2 sin − 1 ( − 7 1 ) ) 61 Question bank Midterm 1443H S1, Q3 A Find the value of cos ( tan − 1 ( − 1 ) + cos − 1 ( 2 3 ) ) \cos\left(\tan^{-1}(-1) + \cos^{-1}\left(\frac{2}{3}\right)\right) cos ( tan − 1 ( − 1 ) + cos − 1 ( 3 2 ) ) without using a calculator. 62 Question bank Midterm 1443H S1, Q3 B If θ \theta θ is in standard position and Q ( − 5 , − 2 ) Q(-5, -2) Q ( − 5 , − 2 ) is on the terminal side of θ \theta θ , find sec θ \sec\theta sec θ . 63 Question bank Midterm 1442H S1, Q3 A Use the reference angle to determine the value of cos ( 315 ∘ ) \cos(315^\circ) cos ( 31 5 ∘ ) . 64 Question bank Midterm 1442H S1, Q3 B Calculate the exact value of cos ( tan − 1 ( 3 5 ) ) \cos\left(\tan^{-1}\left(\frac{3}{5}\right)\right) cos ( tan − 1 ( 5 3 ) ) . 65 Question bank Midterm 1442H S1, Q3 C Verify the identity 1 1 − sin x + 1 1 + sin x = 2 sec 2 x \frac{1}{1 - \sin x} + \frac{1}{1 + \sin x} = 2\sec^2 x 1 − s i n x 1 + 1 + s i n x 1 = 2 sec 2 x . 66 Question bank Midterm 1442H S2, Q3 A Verify the identity ( 1 − sin 2 x ) ( 1 + tan 2 x ) = 1 (1 - \sin^2 x)(1 + \tan^2 x) = 1 ( 1 − sin 2 x ) ( 1 + tan 2 x ) = 1 . 67 Question bank Midterm 1442H S2, Q3 B If Q ( − 5 , − 6 ) Q(-5, -6) Q ( − 5 , − 6 ) is a point on the terminal side of the angle θ \theta θ in standard position, find the values of the six trigonometric functions of θ \theta θ . 68 Question bank Midterm 1442H S2, Q3 C Calculate the exact value of sin ( cos − 1 ( − 2 3 ) ) \sin\left(\cos^{-1}\left(-\frac{2}{3}\right)\right) sin ( cos − 1 ( − 3 2 ) ) . 69 Question bank Alternative Midterm 1443H S1, Q3 A Find the reference angle of θ = 7 6 π \theta = \frac{7}{6}\pi θ = 6 7 π . 70 Question bank Alternative Midterm 1443H S1, Q3 B Show that cos x cos y = 1 2 [ cos ( x + y ) + cos ( x − y ) ] \cos x\cos y = \frac{1}{2}\left[\cos(x + y) + \cos(x - y)\right] cos x cos y = 2 1 [ cos ( x + y ) + cos ( x − y ) ] . 71 Question bank Alternative Midterm 1443H S1, Q3 C Find the exact value of sin ( cos − 1 ( 2 3 ) ) \sin\left(\cos^{-1}\left(\frac{2}{3}\right)\right) sin ( cos − 1 ( 3 2 ) ) . 72 Question bank Midterm 1443H S2, Q3 A Find the exact value of sin ( 2 sin − 1 ( − 3 7 ) ) \sin\left(2\sin^{-1}\left(-\frac{3}{7}\right)\right) sin ( 2 sin − 1 ( − 7 3 ) ) , without using a calculator. 73 Question bank Midterm 1443H S2, Q3 B Prove that tan θ ⋅ sin θ = sec θ − cos θ \tan\theta\cdot\sin\theta = \sec\theta - \cos\theta tan θ ⋅ sin θ = sec θ − cos θ . 74 Question bank Midterm 1438H S1, Q2 C If 3 sec θ + 5 = 0 3\sec\theta + 5 = 0 3 sec θ + 5 = 0 and sin θ > 0 \sin\theta > 0 sin θ > 0 , find sin ( 2 θ ) \sin(2\theta) sin ( 2 θ ) . 75 Question bank Midterm 1438H S2, Q2 C Find the exact value of sin ( 2 cos − 1 ( 3 5 ) ) \sin\left(2\cos^{-1}\left(\frac{3}{5}\right)\right) sin ( 2 cos − 1 ( 5 3 ) ) , without using a calculator. 76 Question bank Alternative Midterm 1438H S2, Q2 C Find the exact value of cos ( 2 cos − 1 ( 4 5 ) ) \cos\left(2\cos^{-1}\left(\frac{4}{5}\right)\right) cos ( 2 cos − 1 ( 5 4 ) ) , without using a calculator. 77 Question bank First Midterm 1439/1440H S1, Q4 A Find the exact value of tan ( cos − 1 ( 5 6 ) ) \tan\left(\cos^{-1}\left(\frac{5}{6}\right)\right) tan ( cos − 1 ( 6 5 ) ) , without using a calculator. 78 Question bank First Midterm 1439/1440H S1, Q4 B Verify the identity sin ( 2 x ) tan x = 2 csc 2 x \sin(2x)\tan x = \frac{2}{\csc^2 x} sin ( 2 x ) tan x = c s c 2 x 2 . 79 Question bank First Midterm 1441H S1, Q4 A Let ϕ \phi ϕ be an angle in standard position, where its rotation is clockwise, with arc length 110 110 110 cm, and the diameter of the circle is 40 40 40 cm. Determine the angle ϕ \phi ϕ in degrees. 80 Question bank First Midterm 1441H S1, Q4 B Find the reference angle of 3 π 4 \frac{3\pi}{4} 4 3 π . 81 Question bank First Midterm 1441H S1, Q4 C Find the exact value of cos ( sin − 1 ( 2 3 ) + tan − 1 ( − 1 3 ) ) \cos\left(\sin^{-1}\left(\frac{2}{3}\right) + \tan^{-1}\left(-\frac{1}{3}\right)\right) cos ( sin − 1 ( 3 2 ) + tan − 1 ( − 3 1 ) ) , without using a calculator. 82 Question bank First Midterm 1441H S1, Q4 D Verify the identity tan 2 x sec 2 x = 1 − cos 2 x \frac{\tan^2 x}{\sec^2 x} = 1 - \cos^2 x s e c 2 x t a n 2 x = 1 − cos 2 x . 83 Question bank First Midterm 1439/1440H S2, Q4 A Find the exact value of cos ( sin − 1 ( 2 3 ) ) \cos\left(\sin^{-1}\left(\frac{2}{3}\right)\right) cos ( sin − 1 ( 3 2 ) ) , without using a calculator. 84 Question bank First Midterm 1439/1440H S2, Q4 B Verify the identity csc 2 x 1 + tan 2 x = cot 2 x \frac{\csc^2 x}{1 + \tan^2 x} = \cot^2 x 1 + t a n 2 x c s c 2 x = cot 2 x . 85 Question bank First Midterm 1441H S2, Q4 A(i) Find the exact value without using a calculator: cos ( 75 ∘ ) \cos(75^\circ) cos ( 7 5 ∘ ) 86 Question bank First Midterm 1441H S2, Q4 A(ii) Find the exact value without using a calculator: sin − 1 ( sin ( 4 π 3 ) ) \sin^{-1}\left(\sin\left(\frac{4\pi}{3}\right)\right) sin − 1 ( sin ( 3 4 π ) ) 87 Question bank First Midterm 1441H S2, Q4 A(iii) Find the exact value without using a calculator: sin ( cos − 1 ( − 1 2 ) + tan − 1 ( 1 ) ) \sin\left(\cos^{-1}\left(-\frac{1}{2}\right) + \tan^{-1}(1)\right) sin ( cos − 1 ( − 2 1 ) + tan − 1 ( 1 ) ) 88 Question bank First Midterm 1441H S2, Q4 B Verify the identity cos ( π 2 − x ) 1 − cos 2 x = csc x \frac{\cos\left(\frac{\pi}{2} - x\right)}{1 - \cos^2 x} = \csc x 1 − c o s 2 x c o s ( 2 π − x ) = csc x . 89 AI generated MCQ Written to fill the MCQ gap 210 ∘ 210^\circ 21 0 ∘ in radians is … \ldots … 90 AI generated MCQ Written to fill the MCQ gap cos ( 5 π 3 ) = … \cos\left(\frac{5\pi}{3}\right) = \ldots cos ( 3 5 π ) = … 91 AI generated MCQ Written to fill the MCQ gap sin − 1 ( sin 5 π 6 ) = … \sin^{-1}\left(\sin\frac{5\pi}{6}\right) = \ldots sin − 1 ( sin 6 5 π ) = … 92 AI generated MCQ Written to fill the MCQ gap If tan θ = − 3 4 \tan\theta = -\frac{3}{4} tan θ = − 4 3 and sin θ > 0 \sin\theta > 0 sin θ > 0 , then cos θ = … \cos\theta = \ldots cos θ = … 93 AI generated MCQ Written to fill the MCQ gap The arc length opposite a central angle of 150 ∘ 150^\circ 15 0 ∘ in a circle of radius 6 6 6 cm is … \ldots … 94 AI generated MCQ Written to fill the MCQ gap cos 2 x ( 1 + tan 2 x ) \cos^2 x\,(1 + \tan^2 x) cos 2 x ( 1 + tan 2 x ) simplifies to … \ldots …