Question

Given that tanθ=−7/4 and π/2<θ<π, find the exact values of the trigonometric functions.
sin2θ=

csc2θ=

cos2θ=

sec2θ=

tan2θ=

cot2θ=

Answer 1

Given that \(\tan \theta = -\frac{7}{4}\) and \(\frac{\pi}{2} < \theta < \pi\), we will find the exact values of the trigonometric functions \( \sin 2\theta\), \( \csc 2\theta\), \( \cos 2\theta\), \( \sec 2\theta\), \( \tan 2\theta\), and \( \cot 2\theta\).

1. **Finding \( \sin \theta \) and \( \cos \theta \)**

We know that:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]
Here, \(\tan \theta = -\frac{7}{4}\), which gives us:
\[
\sin \theta = -7k, \quad \cos \theta = 4k
\]
for some \(k\). Since we know \(\theta\) is in the second quadrant, \(\sin \theta > 0\) and \(\cos \theta < 0\), meaning we need to choose \(k\) such that:
\[
\sin \theta = 7k \quad \text{and} \quad \cos \theta = -4k
\]

We can now use the Pythagorean identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Substituting \( \sin \theta\) and \( \cos \theta \) gives:
\[
(7k)^2 + (-4k)^2 = 1
\]
\[
49k^2 + 16k^2 = 1
\]
\[
65k^2 = 1 \Rightarrow k^2 = \frac{1}{65} \Rightarrow k = \frac{1}{\sqrt{65}} \quad (k > 0)
\]

Now we can find \( \sin \theta\) and \( \cos \theta \):
\[
\sin \theta = 7k = \frac{7}{\sqrt{65}}, \quad \cos \theta = -4k = -\frac{4}{\sqrt{65}}
\]

2. **Finding \( \sin 2\theta\) and \( \cos 2\theta\)**

Using the double angle formulas:
\[
\sin 2\theta = 2 \sin \theta \cos \theta
\]
\[
\sin 2\theta = 2 \left(\frac{7}{\sqrt{65}}\right) \left(-\frac{4}{\sqrt{65}}\right) = -\frac{56}{65}
\]

For \( \cos 2\theta\):
\[
\cos 2\theta = \cos^2 \theta - \sin^2 \theta
\]
\[
\cos^2 \theta = \left(-\frac{4}{\sqrt{65}}\right)^2 = \frac{16}{65}, \quad \sin^2 \theta = \left(\frac{7}{\sqrt{65}}\right)^2 = \frac{49}{65}
\]
Thus:
\[
\cos 2\theta = \frac{16}{65} - \frac{49}{65} = -\frac{33}{65}
\]

3. **Finding \( \csc 2\theta \), \( \sec 2\theta \), \( \tan 2\theta \), and \( \cot 2\theta \)**

Now we compute the rest of the functions using the identities for cosecant, secant, tangent, and cotangent:
\[
\csc 2\theta = \frac{1}{\sin 2\theta} = -\frac{65}{56}
\]

\[
\sec 2\theta = \frac{1}{\cos 2\theta} = -\frac{65}{33}
\]

\[
\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} = \frac{-\frac{56}{65}}{-\frac{33}{65}} = \frac{56}{33}
\]

\[
\cot 2\theta = \frac{1}{\tan 2\theta} = \frac{33}{56}
\]

Putting it all together, we have:

\[
\sin 2\theta = -\frac{56}{65}
\]
\[
\csc 2\theta = -\frac{65}{56}
\]
\[
\cos 2\theta = -\frac{33}{65}
\]
\[
\sec 2\theta = -\frac{65}{33}
\]
\[
\tan 2\theta = \frac{56}{33}
\]
\[
\cot 2\theta = \frac{33}{56}
\]