Mathematics: If tan θ = 3/2 and cos θ 0, use the fundamental identities to evaluate the other five trig functions of θ.

If tan θ = 3/2 and cos θ 0, use the fundamental identities

Ответ:

rambocayden8018

02.01.2022

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$\mathsf{tan\,\theta=\dfrac{3}{2}\qquad\qquad (cos\,\theta$

•   Finding $\mathsf{cos\,\theta:}$

Square both sides of $\mathsf{(i):}$

$\mathsf{(2\,sin\,\theta)^2=(3\,cos\,\theta)^2}\\\\ \mathsf{2^2\,sin^2\,\theta=3^2\,cos^2\,\theta}\\\\ \mathsf{4\,sin^2\,\theta=9\,cos^2\,\theta\qquad\qquad(but~sin^2\,\theta=1-cos^2\,\theta)}\\\\ \mathsf{4\cdot (1-cos^2\,\theta)=9\,cos^2\,\theta}\\\\ \mathsf{4-4\,cos^2\,\theta=9\,cos^2\,\theta}$

$\mathsf{4=9\,cos^2\,\theta+4\,cos^2\,\theta}\\\\ \mathsf{4=13\,cos^2\,\theta}\\\\ \mathsf{cos^2\,\theta=\dfrac{4}{13}}\\\\\\ \mathsf{cos\,\theta=-\,\sqrt{\dfrac{4}{13}}}\qquad\qquad\textsf{(because }\mathsf{cos\,\theta}\textsf{ is negative)}\\\\\\ \mathsf{cos\,\theta=-\,\dfrac{2}{\sqrt{13}}\qquad\quad\checkmark}$

•   Finding $\mathsf{sin\,\theta:}$

Substitute back the value of $\mathsf{cos\,\theta}$ into the equation $\mathsf{(i):}$

$\mathsf{2\,sin\,\theta=3\,cos\,\theta}\\\\ \mathsf{sin\,\theta=\dfrac{3}{2}\,cos\,\theta}\\\\\\ \mathsf{sin\,\theta=\dfrac{3}{\diagup\!\!\!\! 2}\cdot \left(\!-\,\dfrac{\diagup\!\!\!\! 2}{\sqrt{13}} \right)}\\\\\\ \mathsf{sin\,\theta=-\,\dfrac{3}{\sqrt{13}}\qquad\quad\checkmark}$

•   Finding $\mathsf{cot\,\theta:}$

$\mathsf{cot\,\theta=\dfrac{1}{tan\,\theta}}\\\\\\ \mathsf{cot\,\theta=\dfrac{1}{~\frac{3}{2}~}}\\\\\\ \mathsf{cot\,\theta=\dfrac{2}{3}\qquad\quad\checkmark}$

•   Finding $\mathsf{sec\,\theta:}$

$\mathsf{sec\,\theta=\dfrac{1}{cos\,\theta}}\\\\\\ \mathsf{sec\,\theta=\dfrac{1}{-\,\frac{2}{\sqrt{13}}}}\\\\\\ \mathsf{sec\,\theta=-\,\dfrac{\sqrt{13}}{2}\qquad\quad\checkmark}$

•   Finding $\mathsf{csc\,\theta:}$

$\mathsf{csc\,\theta=\dfrac{1}{sin\,\theta}}\\\\\\ \mathsf{csc\,\theta=\dfrac{1}{-\,\frac{3}{\sqrt{13}}}}\\\\\\ \mathsf{csc\,\theta=-\,\dfrac{\sqrt{13}}{3}\qquad\quad\checkmark}$

I hope this helps. =)

Tags: trigonometric trig function sine cosine tangent cotangent secant cosecant sin cos tan cot sec csc relation identity trigonometry

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Ответ:

mairealexander87

03.11.2022

Mathematics

3) In Δ BDC

|DC|/|BC| = cos C

cos C= 16/17.89

C= cos⁻¹( 16/17.89)=26.57⁰

In triangle ABC:

x=180-(90+26.57)=63.43

x=63.43

AB/BD= tan(70⁰), AB=BD*tan(70⁰)

AB/BC=tan(40⁰), AB=BC*tan(40⁰)

BD*tan(70⁰)=BC*tan(40⁰)

BD=BC-CD=BC - 15

(BC -15)*tan(70⁰)=BC*tan(40⁰)

BC*tan(70⁰) -15*tan(70⁰)= BC*tan(40⁰)

BC*tan(70⁰) - BC*tan(40⁰) = 15*tan(70⁰)

BC(tan(70⁰)-tan(40⁰))= 15*tan(70⁰)

BC = 15*tan(70⁰)/(tan(70⁰)-tan(40⁰)) = 21.60

BC=21.60

1) In a quadrilateral sum of angles =360⁰.

PQS=SQR=50⁰, because SQ bisects PQR.

Using Law of sine in ΔSRQ

SR/sin ∠SQR = SQ/sinR, SQ = SR * sinR/sin SQR = 3*sin30/sin50 =3.26

SQ=3.26 cm

ΔPQS:

cos PQS= PQ/SQ

PQ=SQ*cosPQS =3.26*cos 50⁰=2.09=2.1

PQ=2.1 cm

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If tan θ = 3/2 and cos θ < 0, use the fundamental identities to evaluate the other five trig functions of θ.

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