Solve for 𝑡. Round angles to the nearest degree. Give all solutions for 0 ≤ 𝜃 ≤ 360

Question

1. 25 = 50 cos(9𝑡), When 𝜃 = ___________, then 𝑡 = ___________
2. 15 cos(9𝑡) = −10, When 𝜃 = ___________, then 𝑡 = ___________
3. 10 = 5 + 8 cos(20𝑡), When 𝜃 = ___________, then 𝑡 = ___________
4. 120 + 25 cos(0.8𝑡) = 100, When 𝜃 = ___________, then 𝑡 = ___________

Answer 1

To solve these, you need to use inverse trig functions. I'll do the first one for you. You can try the rest.

25 = 50 cos(9𝑡)

1/2 = cos(9t)

cos^-1(1/2) = 9t

(1/9) cos^-1(1/2) = t

The inverse cosine of 1/2 is 60° which you can find using your calculator. You can also find it using a 30-60-90 right triangle. The cosine is positive in quadrants I and IV, so cos^-1(1/2) is also true for 300°. So the answers are (assuming that θ = cos^-1(1/2)):

When θ = 60°, t = 60/9 = 20/3 = 6 ²/₃

When θ = 300°, t = 300/9 = 100/3 = 33 ¹/₃

Answer 2

cos9t = 1/2

9t = 60 degrees or -60 degrees = 300 degrees

theta = 60 or 300 degrees

t = 60/9 = 6.67 degrees or 300/9 = 33.33 degrees

or rounded off to nearest degree: t= 7 or 33 degrees

cos9t = -2/3

9t = 180-48.19 degrees = 131.81 degrees or 180+48.19 = 228.19 degrees

theta = 131.81 or 228.19

t = 131.81/9 - 43.94 degrees or 228.19/9 = 25.35 degrees

rounded off to nearest degree: t = 44 or 25 degrees

cos20t = 5/8

20t = 51.38 or 308.62

theta = 51.38 or 308.62 degrees

t = 51.38/20 = 2.57 degrees or 308.62/20 = 15.43 degrees

rounded off to nearest degree: t=3 or 15 degrees

cos(.8t) =-20/25 = -4/5

0.8t = 180-36.87= 143.13 or 180+36.87 = 216.87 degrees

theta = 143.13 or 216.87 degrees

t = 143.13/.8 or 216.87/.8 = 178.91 or 271.09 degrees

rounded off to nearest degree: t = 179 or 271 degrees

method is first

solve for cos(theta)

then use a scientific calculator with inverse trig functions to solve for theta

then divide by the coefficient of t to solve for t