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,
[tex]\text{ Tan }\theta\text{ = }\frac{y}{x}[/tex][tex]\text{ Cos }\theta\text{ = }\frac{x}{r}[/tex]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:
[tex]a^2+b^2=c^2\text{ }\rightarrow x^2+y^2=r^2[/tex][tex]r\text{ = }\sqrt[]{x^2+y^2}[/tex][tex]\text{ = }\sqrt[]{24^2+7^2}\text{ = }\sqrt[]{576\text{ + 49}}[/tex][tex]\text{ = }\sqrt[]{625}[/tex][tex]r\text{ = 25}[/tex]Let's find the value of Cos θ,
[tex]\text{ Cos }\theta\text{ = }\frac{x}{r}[/tex][tex]\text{ Cos }\theta\text{ = }\frac{24}{25}[/tex]Therefore, the value of Cost θ is 24/25.
SOLUTION 2:
[tex]\text{ Tan }\theta\text{ = }\frac{y}{x}\text{ ; Cos }\theta\text{ = }\frac{x}{r}\text{ ; Sin }\theta\text{ = }\frac{y}{r}[/tex][tex]\text{Tan }\theta\text{ = }\frac{Si\text{n }\theta}{Cos\text{ }\theta}[/tex]Since y = 7 and x = 24, we get:
[tex]\text{Tan }\theta\text{ = }\frac{Si\text{n }\theta}{Cos\text{ }\theta}[/tex][tex]Cos\text{ }\theta\text{ = }\frac{Si\text{n }\theta}{Tan\text{ }\theta}[/tex][tex]Cos\text{ }\theta\text{ = }\frac{\frac{7}{r}}{\frac{7}{24}}\text{ = }\frac{7}{r}x\frac{24}{7}\text{ }[/tex][tex]Cos\text{ }\theta\text{ = }\frac{24}{r}[/tex]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,
[tex]\cos \theta\text{ = }\frac{24}{25}[/tex]
