Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find all angles, 0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree. 4cos^(2)theta=4cos theta-1 Answer: theta=

Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newline4cos⁡2θ=4cos⁡θ−1 4 \cos ^{2} \theta=4 \cos \theta-1 \newlineAnswer: θ= \theta=

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Csc, sec, and cot of special anglesright chevron icon

Full solution

Q. Find all angles, 0∘≤θ<360∘ 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newline4cos⁡2θ=4cos⁡θ−1 4 \cos ^{2} \theta=4 \cos \theta-1 \newlineAnswer: θ= \theta=
  1. Rewrite Equation: Rewrite the given equation in standard quadratic form.\newlineThe given equation is 4cos⁡2(θ)=4cos⁡(θ)−14\cos^2(\theta) = 4\cos(\theta) - 1. To solve for θ\theta, we need to rewrite this equation in the form of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0.\newlineSubtract 4cos⁡(θ)4\cos(\theta) and add 11 to both sides to get:\newline4cos⁡2(θ)−4cos⁡(θ)+1=04\cos^2(\theta) - 4\cos(\theta) + 1 = 0
  2. Factor Quadratic: Factor the quadratic equation.\newlineThe equation 4cos⁡2(θ)−4cos⁡(θ)+1=04\cos^2(\theta) - 4\cos(\theta) + 1 = 0 is a quadratic equation in terms of cos⁡(θ)\cos(\theta). We can factor this equation as:\newline(2cos⁡(θ)−1)2=0(2\cos(\theta) - 1)^2 = 0
  3. Solve for cos⁡(θ)\cos(\theta): Solve for cos⁡(θ)\cos(\theta). Since we have a perfect square, we can take the square root of both sides to get: 2cos⁡(θ)−1=02\cos(\theta) - 1 = 0 Now, add 11 to both sides: 2cos⁡(θ)=12\cos(\theta) = 1 Divide both sides by 22: cos⁡(θ)=12\cos(\theta) = \frac{1}{2}
  4. Find Corresponding Angles: Find the angles that correspond to cos⁡(θ)=12\cos(\theta) = \frac{1}{2}. The cosine function is positive and equals 12\frac{1}{2} at specific angles within the range of 0∘0^\circ to 360∘360^\circ. These angles are 60∘60^\circ and 300∘300^\circ, as cosine is positive in the first and fourth quadrants.
  5. Check Additional Solutions: Check for any additional solutions within the given range.\newlineSince the cosine function is periodic with a period of 360°360°, we need to check if there are any other angles within the range of 0°0° to 360°360° that satisfy the equation. However, since we have already identified the angles where cos⁡(θ)=12\cos(\theta) = \frac{1}{2} within one period, there are no additional solutions.

More problems from Csc, sec, and cot of special angles

Question
Find all solutions with 0∘≤θ≤180∘0^\circ \leq \theta \leq 180^\circ. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.\newlinecos⁡(θ)=−1\cos(\theta) = -1\newline‾∘\underline{\hspace{2em}}^\circ
Get tutor helpright-arrow

Posted 7 months ago

Question
Kimi's school is 1515 kilometers west of her house and 88 kilometers south of her friend Franklin's house. Every day, Kimi bicycles from her house to her school. After school, she bicycles from her school to Franklin's house. Before dinner, she bicycles home on a bike path that goes straight from Franklin's house to her own house. How far does Kimi bicycle each day?\newline_____\_\_\_\_\_ kilometers
Get tutor helpright-arrow

Posted 7 months ago

Question
Evaluate. Write your answer as an integer or as a decimal rounded to the nearest hundredth.\newlinecos⁡35∘=\cos 35^\circ = __
Get tutor helpright-arrow

Posted 7 months ago

Question
Find all solutions with −90∘≤θ≤90∘-90^\circ \leq \theta \leq 90^\circ. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.\newlinecsc⁡(θ)=−1\csc(\theta) = -1\newline____ ∘\_\_\_\_\,^\circ
Get tutor helpright-arrow

Posted 8 months ago

Question
Evaluate. Write your answer in simplified, rationalized form. Do not round.\newlinecot⁡30∘=\cot 30^\circ = ______
Get tutor helpright-arrow

Posted 7 months ago

Question
The terminal side of an angle θ\theta in standard position intersects the unit circle at (8485,1385)(\frac{84}{85}, \frac{13}{85}). What is cos⁡(θ)\cos(\theta)?\newlineWrite your answer in simplified, rationalized form.\newline______
Get tutor helpright-arrow

Posted 7 months ago

Question
−90∘<θ<90∘-90^\circ < \theta < 90^\circ. Find the value of θ\theta in degrees.\newlinetan⁡(θ)=0\tan(\theta) = 0\newlineWrite your answer in simplified, rationalized form. Do not round.\newlineθ=\theta = ____∘^\circ\newline
Get tutor helpright-arrow

Posted 8 months ago

Question
Evaluate. Write your answer in simplified, rationalized form. Do not round.\newlinecsc⁡60∘=\csc 60^\circ = ______
Get tutor helpright-arrow

Posted 8 months ago

Question
θ\theta is an acute angle. Find the value of θ\theta in degrees.\newlinesec⁡(θ)=2\sec(\theta) = \sqrt{2}\newlineθ=\theta = ____°\degree
Get tutor helpright-arrow

Posted 7 months ago

Question
Evaluate. Write your answer in simplified, rationalized form. Do not round.\newlinecos⁡90∘=\cos 90^\circ = ____
Get tutor helpright-arrow

Posted 8 months ago

View all (150)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant