Here. 11) \(2\sin \left ( \dfrac{\pi }{4} \right )2\cos \left ( \dfrac{\pi }{4} \right )\), \(2\sin \left ( \dfrac{\pi }{2} \right )\), 12) \(4\sin \left ( \dfrac{\pi }{8} \right )\cos \left ( \dfrac{\pi }{8} \right )\). The angle to the horizontal is 48 . 2. Our mission is to provide a free, world-class education to anyone, anywhere. 39) A spring attached to the ceiling is pulled \(7\) cm down from equilibrium and released. Explain your reasoning. Find when the spring first comes between \(−0.1\) and \(0.1\) cm, effectively at rest. If the person is \(6\) feet tall, what angle do his feet make with the wall? Verifying a Trigonometric Identity Involving sec2θ. After \(4\) seconds, the amplitude has decreased to \(14\) cm. In this section, we will begin an examination of the … Use the result from part(b) to show that the function \(g(x) = \sin(x) + \sqrt{3}\cos(x)\) is indeed a sinusoidal function. 31) The displacement \(h(t)\) in centimeters of a mass suspended by a spring is modeled by the function \(h(t)=11 \sin (12πt),\) where \(t\) is measured in seconds. Use the Sine Difference Identity to rewrite \(50\sin(120\pi(t - c))\) as an expression of the form \(50\sin(A)\cos(B) - 50\cos(A)\sin(B)\) where \(A\) and \(B\) involve \(t\) and/or \(C\). \(y=6(5)^x+4 \sin \left(\dfrac{π}{2}x \right)\). \[30\sin(120\pi t) + 40\cos(120\pi t) = 50\sin(120\pi(t - c))\].Compare this value of \(C\) to the one you estimated in part (a). Suppose the high temperature of \(105°F\) occurs at 5PM and the average temperature for the day is \(85°F\). This is the currently selected item. (b) We can use \(345^\circ = 300^\circ + 45^\circ\) and first evaluate \(\cos(345^\circ)\). The graph of the sum \(V(t) = V_{1}(t) + V_{2}(t)\) is shown in Figure 4.8. 2) Explain how to determine the double-angle formula for \(\tan(2x)\) using the double-angle formulas for \(\cos(2x)\) and \(\sin (2x)\). Find a function that models the population, \(P\), in terms of months since January, \(t\). The angle made with the horizontal is 57 . If they are the same, show why. 20) Outside temperatures over the course of a day can be modeled as a sinusoidal function. 34) \(\dfrac{1}{1+\cos x}-\dfrac{1}{1-\cos (-x)}=-2\cot x\csc x\), 36) \(\left (\dfrac{\sec ^2(-x)-\tan ^2x}{\tan x} \right )\left (\dfrac{2+2\tan x}{2+2\cot x} \right )-2\sin ^2x=\cos 2x\), 37) \(\dfrac{\tan x}{\sec x}\sin (-x)=\cos ^2x\), 38) \(\dfrac{\sec (-x)}{\tan x+\cot x}=-\sin (-x)\), 39) \(\dfrac{1+\sin x}{\cos x}=\dfrac{\cos x}{1+\sin (-x)}\), Proved with negative and Pythagorean identities. Learn how to solve trigonometric equations and how to use trigonometric identities to solve various problems. 3) After examining the reciprocal identity for \(\sec t\), explain why the function is undefined at certain points. (b) \(x = \dfrac{2\pi}{3} + k(2\pi)\) or \(x = \dfrac{4\pi}{3} + k(2\pi)\), where \(k\) is an integer. Trigonometric identities allow us to simplify a given expression so that it contains sine and cosine ratios only. 27) During a \(90\)-day monsoon season, daily rainfall can be modeled by sinusoidal functions. 26) Find \(\sin \left (\dfrac{\theta }{2} \right )\), \(\cos \left (\dfrac{\theta }{2} \right )\), and \(\tan \left (\dfrac{\theta }{2} \right )\). Use the Pythagorean Identity to obtain \(\sin^{2}(\theta) = \dfrac{5}{9}\). During what period was daily rainfall less than \(5\) inches? 37) A fish population oscillates \(40\) above and below average during the year, reaching the lowest value in January. 27) \(\sin (76^{\circ})+\sin (14^{\circ})\), 28) \(\cos (58^{\circ})-\cos (12^{\circ})\), 29) \(\sin (101^{\circ})-\sin (32^{\circ})\), \(2\cos (66.5^{\circ})\sin (34.5^{\circ})\), 30) \(\cos (100^{\circ})+\cos (200^{\circ})\), 31) \(\sin (-1^{\circ})+\sin (-2^{\circ})\), \(2\sin (-1.5^{\circ})\cos (0.5^{\circ})\). 38) A spring attached to the ceiling is pulled \(10\) cm down from equilibrium and released. \(-2 \sin(-x)\cos(-x)=-2(-\sin(x)\cos(x))=\sin(2x)\), 37) \(\dfrac{1+\cos (2\theta )}{\sin (2\theta )}\tan ^2\theta =\tan \theta\), \(\dfrac{\sin (2\theta )}{1+\cos (2\theta )}\tan ^2\theta =\dfrac{2\sin (\theta )\cos (\theta )}{1+\cos ^2\theta -\sin ^2\theta }\tan ^2\theta=\), \(\dfrac{2\sin (\theta )\cos (\theta )}{2\cos ^2\theta }\tan ^2\theta=\dfrac{\sin (\theta )}{\cos (\theta )}\tan ^2\theta=\), \(\cot (\theta )\tan ^2\theta=\tan \theta\). Floods: July 24 through October 7. Brilliant. The first spring, which oscillates \(14\) times per second, was initially pulled down \(2\) cm from equilibrium, and the amplitude decreases by \(8\%\) each second. Use an appropriate sum or difference identity to find the exact value of each of the following. … Home Algebra II Trigonometric Functions Exercises Trigonometric Identities Exercises . 7) If \(\cos x =-\dfrac{1}{2}\), and \(x\) is in quadrant \(\mathrm{III}\). Secondly, recall that and can be thought of as the x and y coordinates of a particle traversing a unit circle. Does the graph of this function appear to be a sinusoid? If the person is \(5\) feet tall, what angle do her feet make with the wall? A classmate shares his solution to the problem of solving \(\sin(2x) = 2\cos(x)\) over the interval \([0, 2\pi)\). \[\sin(\dfrac{x}{2}) = \sqrt{\dfrac{1 - \dfrac{2}{3}}{2}} = \dfrac{1}{\sqrt{6}}\]. 100) A \(20\)-foot tall building has a shadow that is \(55\) feet long. The key is that the angles are complementary. Use Double Angle Identity to write \(\sin(2A)\) in terms of \(\sin(A)\) and \(\cos(A)\) and to write \(\cos(2A)\) in terms of \(\sin(A)\). A Computer Science portal for geeks. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. 14) \(\csc \left (\dfrac{\pi }{2}-t \right)\), 15) \(\sec \left (\dfrac{\pi }{2}-\theta \right)\), 16) \(\cot \left (\dfrac{\pi }{2}-x \right)\), 17) \(\tan \left (\dfrac{\pi }{2}-x \right)\), 18) \(\sin(2x)\cos(5x)-\sin(5x)\cos(2x)\), 19) \(\dfrac{\tan \left (\dfrac{3}{2}x \right)-\tan \left (\dfrac{7}{5}x \right)}{1+\tan \left (\dfrac{3}{2}x \right)\tan \left (\dfrac{7}{5}x \right)}\). If she is standing \(4\) miles from the launch pad, at what angle is she looking up from horizontal? For the exercises 18-29, construct a sinusoidal function with the provided information, and then solve the equation for the requested values. Use the quadratic formula if the equations do not factor. For the exercises 14-16, graph the given function, and then find a possible physical process that the equation could model. Consider “rest” as an amplitude less than \(0.1\) cm. It is often helpful to use the definitions to rewrite all trigonometric functions in terms of the cosine and sine. Solution for 1.3 Evaluating Trigonometric Functions In Exercises 33 and 34, find the exact values of the six trigonometric functions of the angle 0. Verify those equations that are identities and provide examples to show that the others are not identities. Exercise 2 Knowing that $tan { alpha }… For the exercises 42-46, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point. A low of \(4\) inches of rainfall was recorded on day \(30\), and overall the average daily rainfall was \(8\) inches. Suppose the temperature varies between \(64°F\) and \(86°F\) during the day and the average daily temperature first occurs at 12 AM. For the exercises 54-55, prove the following sum-to-product formulas. For the exercises 14-19, simplify the given expression. If a man is standing \(2\) miles from the launch pad, at what angle is she looking down at him from horizontal? The basic idea behind X-ray crystallography is this: two X-ray beams with the same wavelength \(\lambda\) and phase are directed at an angle \(\theta\) toward a crystal composed of atoms arranged in a lattice in planes separated by a distance \(d\) as illustrated in Figure4.5.1 The beams reflect off different atoms (labeled as \(P\) and \(Q\) in Figure 4.5) within the crystal. 93) An airplane has only enough gas to fly to a city \(200\) miles northeast of its current location. \(\cos(-10^\circ - 35^\circ) = \cos(-45^\circ) = \dfrac{\sqrt{2}}{2}\). How can we tell whether the function is even or odd by only observing the graph of \(f(x)=\sec x\)? The second spring, oscillating \(22\) times per second, was initially pulled down \(10\) cm from equilibrium and after \(3\) seconds has an amplitude of \(2\) cm. 4) \(\cos \left (\dfrac{7\pi }{12} \right)\), 5) \(\cos \left (\dfrac{\pi }{12} \right)\), 6) \(\sin \left (\dfrac{5\pi }{12} \right)\), 7) \(\sin \left (\dfrac{11\pi }{12} \right)\), 8) \(\tan \left (-\dfrac{\pi }{12} \right)\), 9) \(\tan \left (\dfrac{19\pi }{12} \right)\), For the exercises 10-13, rewrite in terms of \(\sin x\) and \(\cos x\), 10) \(\sin \left (x+\dfrac{11\pi }{6} \right)\), 11) \(\sin \left (x-\dfrac{3\pi }{4} \right)\), \(-\dfrac{\sqrt{2}}{2}\sin x-\dfrac{\sqrt{2}}{2}\cos x\), 12) \(\cos \left (x-\dfrac{5\pi }{6} \right)\), 13) \(\cos \left (x+\dfrac{2\pi }{3} \right)\), \(-\dfrac{1}{2}\cos x-\dfrac{\sqrt{3}}{2}\sin x\). (Hint: examine the values of \(\cos x\) necessary for the denominator to be \(0\).). Solved Exercises. 22) \(\sin \left ( \cos^{-1}\left ( 0 \right )- \cos^{-1}\left ( \dfrac{1}{2} \right )\right )\), 23) \(\cos \left ( \cos^{-1}\left ( \dfrac{\sqrt{2}}{2} \right )+ \sin^{-1}\left ( \dfrac{\sqrt{3}}{2} \right )\right )\), 24) \(\tan \left ( \sin^{-1}\left ( \dfrac{1}{2} \right )- \cos^{-1}\left ( \dfrac{1}{2} \right )\right )\). Find all solutions to the given equation. (Hint: this is called the angle of depression.). \(\dfrac{3+\cos(4x)-4\cos(2x)}{3+\cos(4x)+4\cos(2x)}\), \(\dfrac{3+\cos(4x)-4\cos(2x)}{4(\cos(2x)+1)}\), 52) \(\tan ^2\left ( \dfrac{x}{2} \right )\sin x\). A student is asked to approximate all solutions in degrees (to two decimal places) to the equation \(\sin(\theta) + \dfrac{1}{3} = 1\) on the interval \(0^\circ \leq \theta \leq 360^\circ\).The student provides the answer \(\theta = \sin^{-1}(\dfrac{2}{3}) \approx 41.81^\circ\). What is the period of this sinusoid. Exercise 3.7: Trigonometric Identities: Product to Sum and Sum to Product Identities. The parentheses around the argument of the functions are often omitted, e.g., sin θ and cos θ, if an interpretation is unambiguously possible. How to use trig identities to rewrite trig expressions? (a) This is an identity. Khan Academy is a 501(c)(3) nonprofit organization. Using Half-Angle Identities to Solve a Trigonometric Equation Find the exact value of the following: sin (22.5°) … 23)A Ferris wheel is \(45\) meters in diameter and boarded from a platform that is \(1\) meter above the ground. (1 – cos 2 A) cosec 2 A = 1 Solution: (1 – cos 2 A) cosec 2 A = 1 L.H.S. Adopted a LibreTexts for your class? 17) A city’s average yearly rainfall is currently \(20\) inches and varies seasonally by \(5\) inches. Then use the values of \(\sin(\dfrac{\pi}{3})\) and \(\cos(\dfrac{\pi}{3})\) to further rewrite the expression. What is the angle of elevation of the sun? Note, the model is invalid once it predicts negative rainfall, so choose the first point at which it goes below \(0\). For the exercises 38-44, rewrite the expression with an exponent no higher than 1. Give examples of two different sets of information that would enable modeling with an equation. Find a function that models the population, \(P\), in terms of months since January, \(t\). Fundamental Identities. For the exercises 56-63, prove the identity. Does the graph of this function appear to be a sinusoid? For the exercises 27-31, rewrite the sum as a product of two functions. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. \[\cos^{2}(x) + 4\sin(x) = 4\] 40) A spring attached to the ceiling is pulled \(17\) cm down from equilibrium and released. Draw graphs to determine if a given equation is an identity. Attempt this array of worksheets encompassing exercises to find the value of trigonometric functions based on the given trig ratios and the quadrant. Trigonometry (10th Edition) answers to Chapter 5 - Trigonometric Identities - Section 5.2 Verifying Trigonometric Identities - 5.2 Exercises - Page 202 2 including work step by step written by community members like you. 1) Explain the basis for the cofunction identities and when they apply. For the exercises 53-54, find a function of the form \(y=ab^x \cos \left(\dfrac{π}{2}x \right)+c\) that fits the given data. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview … 25) The sea ice area around the South Pole fluctuates between about \(18\) million square kilometers in September to \(3\) million square kilometers in March. Brilliant. If the equation appears to be an identity, prove the identity. Explain why or why not. Using the identities: tanθ ≡sinθ/cosθ and sin²θ+cos²θ ≡1; Quadrant rule to solve trig equations Sum to Product and Product to Sum Identities. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. See Trigonometric Graphs. 21) If \(\sin x =-\dfrac{12}{13}\), and \(x\) is in quadrant \(\mathrm{III}\). 34) A deer population oscillates \(19\) above and below average during the year, reaching the lowest value in January. \(\dfrac{120}{169}, -\dfrac{119}{169}, -\dfrac{120}{119}\). For example, \(\sin^{2}(x) + \sin(x)\) can be written as \(\sin(x)(\sin(x) + 1)\) and \(\dfrac{1}{\sin(x)} + \dfrac{1}{\cos(x)}\) can be written as the single fraction \(\dfrac{\cos(x) + \sin(x)}{\sin(x)\cos(x)}\) with a common denominator.