If θ is an acute angle and 5 𝑐𝑜𝑠𝑒𝑐 θ = 7, then evaluate 𝑠𝑖𝑛θ + 𝑐𝑜𝑠2θ − 1.

Solution:

We have been given that 5 𝑐𝑜𝑠𝑒𝑐 θ = 7, where θ is an acute angle and we have to find the value of 𝑠𝑖𝑛θ + 𝑐𝑜𝑠²θ − 1.

$\begin{aligned} 5 cosec θ = 7 \\ \Rightarrow cosec θ = \frac{7}{5} \end{aligned}$

We know that 𝑐𝑜𝑠𝑒𝑐 θ = 1 / 𝑠𝑖𝑛θ

Therefore, 𝑠𝑖𝑛 θ = 5/7

We also know that $\sin^{2} θ + \cos^{2} θ = 1$

$\begin{array}{l} \Rightarrow \cos^{2} θ = 11 - \sin^{2} θ \\ \Rightarrow \cos^{2} θ = 1 - {(\frac{5}{7})}^{2} \\ \cos^{2} θ = \frac{24}{49} \end{array}$

Substituting these values in the expression 𝑠𝑖𝑛θ + 𝑐𝑜𝑠² θ, we get

$\begin{array}{ll} \sin & θ + \cos^{2} θ - 1 = \frac{5}{7} + \frac{24}{49} - 1 = \frac{10}{49} \end{array}$

Therefore the value of the given expression is $\frac{10}{49}$ .

If θ is an acute angle and 5 𝑐𝑜𝑠𝑒𝑐 θ = 7, then evaluate 𝑠𝑖𝑛θ + 𝑐𝑜𝑠²θ − 1.

Solution:

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