the value of one trigonometric function of an acute angle 3 , find the volues of the ramaining five trigonometric functions of . Gue the eract answar as a olfied fraction, if necessary, Rationalice denominators if necescary. $$ \cos 5=\frac{\sqrt{5}}{5} $$

Question

UpStudy AI · Accepted Answer

Given that $$\cos \theta = \frac{\sqrt{5}}{5}$$ for an acute angle $$\theta$$, we can find the remaining five trigonometric functions as follows:

1. **Sine ($$\sin \theta$$)**:
   $$
   \sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left(\frac{\sqrt{5}}{5}\right)^2} = \sqrt{1 - \frac{5}{25}} = \sqrt{\frac{20}{25}} = \frac{2\sqrt{5}}{5}
   $$

2. **Tangent ($$\tan \theta$$)**:
   $$
   \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{2\sqrt{5}}{5}}{\frac{\sqrt{5}}{5}} = 2
   $$

3. **Cotangent ($$\cot \theta$$)**:
   $$
   \cot \theta = \frac{1}{\tan \theta} = \frac{1}{2}
   $$

4. **Secant ($$\sec \theta$$)**:
   $$
   \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{\sqrt{5}}{5}} = \frac{5}{\sqrt{5}} = \sqrt{5}
   $$

5. **Cosecant ($$\csc \theta$$)**:
   $$
   \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{2\sqrt{5}}{5}} = \frac{5}{2\sqrt{5}} = \frac{\sqrt{5}}{2}
   $$

**Summary of All Six Trigonometric Functions:**

$$
\begin{aligned}
\sin \theta &= \frac{2\sqrt{5}}{5} \
\cos \theta &= \frac{\sqrt{5}}{5} \
\tan \theta &= 2 \
\cot \theta &= \frac{1}{2} \
\sec \theta &= \sqrt{5} \
\csc \theta &= \frac{\sqrt{5}}{2} \
\end{aligned}
$$
$$ \begin{aligned} \sin \theta &= \frac{2\sqrt{5}}{5} \ \cos \theta &= \frac{\sqrt{5}}{5} \ \tan \theta &= 2 \ \cot \theta &= \frac{1}{2} \ \sec \theta &= \sqrt{5} \ \csc \theta &= \frac{\sqrt{5}}{2} \ \end{aligned} $$

the value of one trigonometric function of an acute angle 3 , find the volues of the ramaining five trigonometric functions of . Gue the eract answar as a olfied fraction, if necessary, Rationalice denominators if necescary.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Trigonometry Questions