Use a calculator to solve the equation on the interval [0 , 2π]. 2 tan²(x) – 7 tan(x) + 5 =0

Question

Use a calculator to solve the equation on the interval [0 , 2π].
2 tan²(x) – 7 tan(x) + 5 =0

in progress 0
Delilah 2 weeks 2021-10-01T06:02:42+00:00 1 Answer 0

Answers ( )

    0
    2021-10-01T06:04:23+00:00

    Answer:

    \theta_{1} \approx 0.379\cdot \pi or \theta_{1} \approx 1.379\cdot \pi, \theta_{2} = 0.25\cdot \pi or \theta_{2} = 1.25\cdot \pi

    Step-by-step explanation:

    Let use the following substitution formula:

    u = \tan \theta

    The trigonometric expression is converted into an algebraic one, a second-order polynomial:

    2\cdot u^{2}-7\cdot u + 5 = 0

    Roots can be found by using the General Equation for Second-Order Polynomial:

    u = \frac{7\pm \sqrt{49 - 40} }{4}

    Roots are u_{1} = 2.5 and u_{2} = 1. As tangent function has a periodicity of \pi, solutions of u_{1} and u_{2} belong to first and third quadrants. Then, angles can be easily found by using inverse trigonometric functions:

    \theta_{1} = \tan^{-1} u_{1}

    \theta_{1} \approx 0.379\cdot \pi or \theta_{1} \approx 1.379\cdot \pi

    \theta_{2} = \tan^{-1} u_{2}

    \theta_{2} = 0.25\cdot \pi or \theta_{2} = 1.25\cdot \pi

Leave an answer

45:7+7-4:2-5:5*4+35:2 =? ( )