given that tan(t) = 5/12, and 0 < t < π/2, co...
given that tan(t) = 5/12, and 0 < t < π/2, complete the steps to find cos(t). which identity would be best to start with? sin²(t) + cos²(t) = 1 tan²(t) + 1 = sec²(t) 1 + cot²(t) = csc²(t)
Answer
# Explanation:
## Step1: Recall the relationship between tangent and secant
We know that $\tan(t)=\frac{5}{12}$, and the identity $\tan^{2}(t)+1 = \sec^{2}(t)$ is relevant as it directly involves the tangent - function which we know the value of. Substitute $\tan(t)=\frac{5}{12}$ into the identity:
$\left(\frac{5}{12}\right)^{2}+1=\sec^{2}(t)$
## Step2: Calculate the value of $\sec^{2}(t)$
First, calculate $\left(\frac{5}{12}\right)^{2}=\frac{25}{144}$. Then $\frac{25}{144}+1=\frac{25 + 144}{144}=\frac{169}{144}$, so $\sec^{2}(t)=\frac{169}{144}$.
## Step3: Find the value of $\sec(t)$
Since $0\lt t\lt\frac{\pi}{2}$, $\sec(t)>0$. Taking the square - root of $\sec^{2}(t)=\frac{169}{144}$, we get $\sec(t)=\frac{13}{12}$.
## Step4: Recall the relationship between secant and cosine
We know that $\sec(t)=\frac{1}{\cos(t)}$. Since $\sec(t)=\frac{13}{12}$, then $\cos(t)=\frac{12}{13}$.
# Answer:
$\frac{12}{13}$