Trigonometric equations Questions and Answers

A geologist wants to measure the diameter of a crater. From her camp, it is 5 miles to the northernmost point of the crater and 3 miles to the southernmost point. If the angle between the two lines of sight is 112°, what is the diameter of the crater? Round your answer to the nearest tenth of a mile. ____________mile.

Express the given sum as a product of sines and/or cosines. cosx/2 + cos19x/2

Solve these equations algebraically. Find all solutions of each equation on the interval (0,2π). Give exact answers when possible. Round approximate answers to the nearest hundredth. 4 sin^2 x - sin x + 1 = 2 cos^2x

IF 0 = arccos(-2/3), then (a)cosθ=-2/3 and -3π/2< θ < -π

In which quadrants is tan x = sin x/√(1-sin² x)? (A) I, II (B) I, III (C)I, IV (D) All Quadrants

Find all the square roots of the complex number -3 - 2i. Write the square roots in trigonometric form, r(cosθ+ i sin θ), with the smaller angle first. Give your angles in degrees rounded to 4 places, but do not use a degree symbol. Root #1: Root #2: Write the square roots in a + bi form, with the smaller angle first: Round to two decimal places. Root #1: Root #2:

Evaluate the expression. sin²240° - cos ²180° + tan ²60°=____ (Simplify your answer, including any radicals. Use integers

If √-4+√8+16 cosec θ + sin θ = (5√3)/6 , where θ lies in 1st quadrant, then tanθ/2 equals (1) √6 (2) √3 (3)1/(2√3) (4)1/√3

Find all the complex cube roots of w = 8( cos 210° + i sin 210°). Write the roots in polar form with 0 in degrees.

Write the following complex number in rectangular form. 14( cos 135°+ i sin 135°)

Sketch a graph that shows at least one full period. Fill in the table of the "five important points" using the method shown in the notes: y=-1+cos [2(x-π/4)]

Convert the polar equation r² (25cos²θ + 4sin²θ) =100 to a rectangular equation.

Use the given conditions. cos(u) = 5/13, 0<u<π/2 (a) Determine the quadrant in which u/2 lies. Quadrant I Quadrant II Quadrant III Quadrant IV (b) Find the exact values of sin(u/2), cos(u/2), and tan(u/2) using the half-angle formulas. sin(u/2)= cos(u/2) = tan(u/2)=

The function graphed to the right is of the form y = a sec bx + c or y = a csc bx + c for some a ≠ 0, b>0. Determine the equation of the function. An equation of the function shown is y =

Given the equation y = 3+2sin(2x-π) Identify the following: 1) amplitude, 2) period 3) phase shift and 4) give the coordinates of at least 5 key points. Amplitude = Period =

The domain of both the sine and cosine functions is ___, and the range is ____. (Type your answers in interval notation. Type exact answers, using π as needed.)

Convert the Cartesian coordinate (-3,-3) to polar coordinates, 0≤ θ < 2π, r> 0 r= Enter exact value. θ=

Find all real numbers that satisfy the equation. Round approximate answers to the nearest hundredth. 2 sin(πx) = Group of answer choices {x|x = 0.33 + 2k or x = 0.67 + 2k} {x|x = 0.33 + k or x = 1.67 + k} {x|x = 0.17 + 2k or x = 0.83 + 2k} {x|x = 0.33 + k or x = 0.67 + k}

[11] Solve for the variable over C. Circle answers in r cis θ form. -168 + 3x³ = 2019

Find all solutions of the equation in the interval [0, 2π). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) 4 cos x + 4 sin x tan x = 8 X= Find all solutions of the equation in the interval [0, 2r). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) 5 sec x + 5 tan x = 5 X=

An equivalent expression to cos(θ ) tan(θ ) is: a) sin(θ) b)1/cos (θ ) c) 2 sin(θ) cos(θ) d) tan(θ )

For which of the following is there no solution for x? a) sin(x) - 3 = 0 b) sin(x) = 0.5 c) sin(x) = -0.5 d) sin(x) - 1 = 0

A function of the form f(x) = cos(x) + c passes through the point A(π/3, 1). The function is: a) f(x) = cos(x) +0.5 b) f(x) = cos(x) + 1 c) f(x) = cos(x) + 2π/3 d) f(x) = cos(x) – 0.5

In a small fishing village outside of St. John's, Newfoundland, the tides can be modeled by the function: H(t) = 2.5 cos 2π(t - 3)/12 + 4 With H(t) measured in metres and t measured in hours after midnight. When will the tide height be 2.5 m (for the first time)? a) just after 7 am b) 16.3 days c) just after 7 days d) the tide will never be this height

To calculate the height of a tower Mary measures the angle of elevation from a point A, to be 100. She then walks directly towards the tower, and finds the angle of elevation from the new point B to be 200. What is the height of the tower to the nearest tenth of a meter? Show all your work.

Amy is flying a plane at an altitude of 1150 feet and sees a cornfield at an angle of depression of 30°. What is the plane's approximate horizontal distance to the cornfield at this point?

Prove the following trig identities. Show all of your steps for full marks. a) sec θcos θ + sec θ sinθ = 1 + tan θ b) sin² x +sin x cos x/tan x = 1

Using the Law of Sines to solve the all possible triangles if A - If no answer exists, enter DNE for all answers. ZB is 31 x degrees ZC is 29 x degrees C = 18 120°, a 32, 6 = 19. Assume A is opposite side a, ZB is opposite side b, and ≤0 is opposite side c.

Prove the identity.=-cscx - sinx = cotx cosx

Use the product-to-sum identities to rewrite the following expression as a sum or difference. 4sin (3π/4) cos (3π/4)

Use the sum and difference identities to determine the exact value of the following expression. If the answer is undefined, write DNE. sin 165 cos75 - cos165 sin75

for sin 2x + Cas x = 0, Use a double angle or half angle formula to simplify the equation and then find all solutions of equation in interval [0, 2π). x₁ = x2= X3²

Solve sin(4x) cos(9x) - cos(6x) sin(9x) = -0.45 for the smallest positive solution. X =

Solve 5 sin(2x) +1 cos(x) =D for all solutions 0≤x≤ 2π

Q: The range of the function f(x) = log√13, (5 sin ( (π/3)+ x) +12 sin ((5π/6)+x))) is a) (0,2] b) (-∞, 2] c) (-∞, 1/2] d) (0, 1/2]

A solution of the first order differential equation y'cos(x+y)+ sin(x+y)/x=e^x-cos(x + y) is (a) sin(x + y)- e^x = constant (b) e^xxtan(x + y) = constant (c) x(cos(x + y)-e^x) + e^x = constant (d) x(sin(x + y) -e^x) + e^x = constant

Solve for t, 0 < t < 2π. 27 sin(t) cos(t) = -9 cos(t) I

Solve 2 cos² (w) + cos(w) - 1 = 0 for all solutions 0 ≤w < 2. < W= Give your answers as exact values in a list separated by commas.

Graph two periods of the given tangent function. (x- π / 14 ) y = tan Choose the correct graph of two periods of y= tan ( x- π / 14 )

A boat leaves the entrance to a harbor and travels 175 miles on a bearing of N 54° E. How many miles north and how many miles east from the harbor has the boat traveled?

Use the trigonometric function values of the quadrantal angles to evaluate (sin 90°)² +8( tan 0°)² +5 cos 180° (sin 90°)² + 8( tan 0°)² +5 cos 180° = (Simplify your answer, including any radicals. Use integers or fractions for a

Find all solutions to the equation. 4sint-4cost=4 Find all solutions to the equation. 4sin ²t+2sint=2

Verify the identity. sec ² θ(1-sin ² θ) = 1 To verify the identity, start with the more complicated side and transform it to look like the other side. Choose the correct transformations and transform the expression at each step.

The bearing from A to C is S61° E The bearing from A to B is N 87° E. The bearing from B to C is S 29° W A plane flying at 250 mph takes 3 4 hr to go from A to B. Find the distance from A to C The distance from A to C is mi (Round to the nearest integer as needed.)

Rewrite sin (x +7π/4) in terms of sin(x) and cos(x).

The trend of covid cases in Ontario seems to be a neverending sinusoidal function of ups and downs. If the trend eventually becomes the seasonal flu over a 12-month period, with a minimum number impacted in August of 100 cases. Create an equation of such a cosine function that will ensure the minimum number of cases is 100. Note that the maximum cases can be any reasonable value of your choice. Assume 0 = December, 1 = January, 2 = February and so on.

Simplify sin² (t) + cos² (t) / cos² (t) to an expression involving a single trig function with no fractions.

2. Test for symmetry and graph the following equation: r=3 sin 2θ/sin³ θ + cos^3 θ

7. For the given function, state the amplitude and the maximum output for the function: f(t) = 2sin(t) The amplitude is 4 and the maximum value is y = 2. The amplitude is 2 and the maximum value is y = 4. The amplitude is 2 and the maximum value is y = 2. The amplitude is 4 and the maximum value is y = 4.

Give the exact value of the expression without using a calculator. cos (tan-¹(-8))