Home > Trigonometry > Trigonometry

Express (picture below) in terms of sin θ?

Please show work and/or explain! :]

Answer 1

Charles Autterson

#sin^2theta#

#"let's begin by simplifying the contents of the radical"#

#"using the "color(blue)"trigonometric identities"#

#•color(white)(x)tantheta=sintheta/costheta" and "cottheta=costheta/sintheta#

#•color(white)(x)sectheta=1/costheta" and "sin^2theta+cos^2theta=1#

#rArr(tan^2theta)/(sec^2theta+cot^2thetasec^2theta)#

#=(sin^2theta/cos^2theta)/(1/cos^2theta+cancel(cos^2theta)/sin^2thetaxx1/cancel(cos^2theta)#

#=(sin^2theta/cos^2theta)/(1/cos^2theta+1/sin^2theta#

#=(sin^2theta/cos^2theta)/((sin^2theta+cos^2theta)/(cos^2thetasin^2theta)#

#=(sin^2theta/cos^2theta)/(1/(cos^2thetasin^2theta)#

#=sin^2theta/cancel(cos^2theta)xxcancel(cos^2theta)sin^2theta#

#=sin^4theta#

#rArrsqrt((tan^2theta)/(sec^2theta+cot^2thetasec^2theta)#

#=sqrt(sin^4theta)=sin^2theta#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Elijah Alexander

#sqrt ((tan^2 θ) / (sec^2 θ + cot^2 θ sec^2 θ)) = sin^2 θ#

where #θ ≠ kpi + pi/2, kpi - pi/4#, k any integer

#sqrt ((tan^2 θ) / (sec^2 θ + cot^2 θ sec^2 θ))#

#sqrt ((sin^2 θ/cos ^2 θ) / (sec^2 θ( 1 + cot^2 θ))#

#sqrt ((sin^2 θ/cos ^2 θ) / (1/cos^2 θ( 1 + cot^2 θ))#

#sqrt ((sin^2 θ) / (( 1 + cot^2 θ))#

#=sqrt (sin^2 θ / ( ( 1 + (1/sin^2θ - 1 )#

# = sqrt (sin^2 θ / (1/sin^2 θ )) #

# = sqrt (sin^4 θ ) #

# = sin^2 θ # , where #cos ^2 θ and 1 + cot^2 θ≠0#

#cos ^2 θ = 0, θ = kpi + pi/2#, k any integer #1 + cot^2 θ = 0, θ = kpi - pi/4#, k any integer

Note: How #cot^2 θ # became # 1/sin^2θ - 1# in the third line:

#sin^2 θ+ cos^2 θ = 1# #sin^2 θ/sin^2 θ+ cos^2 θ/sin^2 θ = 1/sin^2 θ# #1 + cot^2θ = 1/sin^2 θ# #cot^2θ = 1/sin^2 θ - 1#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 3

Kai Anderson

The expression in the picture can be expressed in terms of sin θ as follows:

cos(θ) / sin(θ) = cot(θ)

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.