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. -7sin^(2)theta-4sin theta+2=-1 Answer: theta=

Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newline−7sin⁡2θ−4sin⁡θ+2=−1 -7 \sin ^{2} \theta-4 \sin \theta+2=-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.\newline−7sin⁡2θ−4sin⁡θ+2=−1 -7 \sin ^{2} \theta-4 \sin \theta+2=-1 \newlineAnswer: θ= \theta=
  1. Simplify Equation: Simplify the given equation by adding 11 to both sides to set the equation to zero.\newline−7sin⁡2θ−4sin⁡θ+2=−1-7\sin^{2}\theta - 4\sin \theta + 2 = -1\newline−7sin⁡2θ−4sin⁡θ+2+1=0-7\sin^{2}\theta - 4\sin \theta + 2 + 1 = 0\newline−7sin⁡2θ−4sin⁡θ+3=0-7\sin^{2}\theta - 4\sin \theta + 3 = 0
  2. Factor Quadratic Equation: Factor the quadratic equation in terms of sin⁡(θ)\sin(\theta). We are looking for two numbers that multiply to −7×3=−21-7 \times 3 = -21 and add up to −4-4. These numbers are −7-7 and +3+3. So, we can write the equation as: (−7sin⁡(θ)+3)(sin⁡(θ)−1)=0(-7\sin(\theta) + 3)(\sin(\theta) - 1) = 0
  3. Solve for sin⁡(θ)\sin(\theta): Solve each factor set to zero for sin⁡(θ)\sin(\theta).\newlineFirst factor: −7sin⁡(θ)+3=0-7\sin(\theta) + 3 = 0\newline−7sin⁡(θ)=−3-7\sin(\theta) = -3\newlinesin⁡(θ)=37\sin(\theta) = \frac{3}{7}\newlineSecond factor: sin⁡(θ)−1=0\sin(\theta) - 1 = 0\newlinesin⁡(θ)=1\sin(\theta) = 1
  4. Find Corresponding Angles: Find the angles that correspond to sin⁡(θ)=37\sin(\theta) = \frac{3}{7} and sin⁡(θ)=1\sin(\theta) = 1 within the range 0^\circ \leq \theta < 360^\circ. For sin⁡(θ)=1\sin(\theta) = 1, the angle is 90∘90^\circ because sin⁡(90∘)=1\sin(90^\circ) = 1. For sin⁡(θ)=37\sin(\theta) = \frac{3}{7}, we need to use the inverse sine function or a calculator to find the angles. Using a calculator, we find: θ=arcsin⁡(37)≈25.4∘\theta = \arcsin(\frac{3}{7}) \approx 25.4^\circ Since the sine function is positive in the first and second quadrants, we also need to find the angle in the second quadrant. θ=180∘−25.4∘≈154.6∘\theta = 180^\circ - 25.4^\circ \approx 154.6^\circ
  5. Verify Solutions: Verify that the angles found are within the given range and list all the solutions.\newlineThe angles 90∘90^\circ, 25.4∘25.4^\circ, and 154.6∘154.6^\circ are all within the range 0^\circ \leq \theta < 360^\circ.\newlineTherefore, the solutions are θ≈25.4∘\theta \approx 25.4^\circ, 90∘90^\circ, and 154.6∘154.6^\circ.

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 9 months ago

View all (150)

Related topics

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