Question

Given tan θ= 7/24 and θ terminates in quadrant 3 what is the value of cos θ

Answer 1

Given the following:

a.) tan θ = 7/24

Let's make a graph to better understand the problem,

SOLUTION 1:

Let's first recall the function of Tan θ and Cos θ,

`\text{ Tan }\theta\text{ = }(y)/(x)`
`\text{ Cos }\theta\text{ = }(x)/(r)`

From the given function, it appears that x = 24 and y = 7. To be able to determine the value of Cos θ, we must first determine the value of r using the Pythagorean Theorem:

`a^2+b^2=c^2\text{ }\rightarrow x^2+y^2=r^2`
`r\text{ = }\sqrt[]{x^2+y^2}`
`\text{ = }\sqrt[]{24^2+7^2}\text{ = }\sqrt[]{576\text{ + 49}}`
`\text{ = }\sqrt[]{625}`
`r\text{ = 25}`

Let's find the value of Cos θ,

`\text{ Cos }\theta\text{ = }(x)/(r)`
`\text{ Cos }\theta\text{ = }(24)/(25)`

Therefore, the value of Cost θ​ is 24/25.

SOLUTION 2:

`\text{ Tan }\theta\text{ = }(y)/(x)\text{ ; Cos }\theta\text{ = }(x)/(r)\text{ ; Sin }\theta\text{ = }(y)/(r)`
`\text{Tan }\theta\text{ = }\frac{Si\text{n }\theta}{Cos\text{ }\theta}`

Since y = 7 and x = 24, we get:

`\text{Tan }\theta\text{ = }\frac{Si\text{n }\theta}{Cos\text{ }\theta}`
`Cos\text{ }\theta\text{ = }\frac{Si\text{n }\theta}{Tan\text{ }\theta}`
`Cos\text{ }\theta\text{ = }((7)/(r))/((7)/(24))\text{ = }(7)/(r)x(24)/(7)\text{ }`
`Cos\text{ }\theta\text{ = }(24)/(r)`

From the given graph, we found out that r = 25 after applying the Pythagorean Theorem. Therefore, to complete the value of cos θ,

​we substitute r = 25.

Therefore,

`\cos \theta​\text{ = }(24)/(25)`