Sure, let's break down each part of the given problem.
### a) [tex]\(\sin 77^\circ\)[/tex]
To express [tex]\(\sin 77^\circ\)[/tex] in terms of [tex]\(b\)[/tex], recall the complementary angle identity:
[tex]\[ \sin \theta = \cos (90^\circ - \theta) \][/tex]
For [tex]\(\theta = 77^\circ\)[/tex]:
[tex]\[ \sin 77^\circ = \cos (90^\circ - 77^\circ) = \cos 13^\circ \][/tex]
Given that [tex]\(\cos 77^\circ = b\)[/tex], we know that:
[tex]\[ \cos 13^\circ = b \][/tex]
So,
[tex]\[ \sin 77^\circ = b \][/tex]
Thus, the value is:
[tex]\[ \sin 77^\circ \approx 0.9743700647852352 \][/tex]
### b) [tex]\(\tan 13^\circ\)[/tex]
Using the definition of [tex]\(\tan \theta\)[/tex], we have:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
For [tex]\(\theta = 13^\circ\)[/tex]:
[tex]\[ \tan 13^\circ = \frac{\sin 13^\circ}{\cos 13^\circ} \][/tex]
We already know [tex]\(\cos 13^\circ = b\)[/tex], and we can use [tex]\(\sin 13^\circ\)[/tex]:
[tex]\[ \sin 13^\circ = \sqrt{1 - (\cos 13^\circ)^2} = \sqrt{1 - b^2} \][/tex]
So,
[tex]\[ \tan 13^\circ = \frac{\sqrt{1 - b^2}}{b} \][/tex]
Thus, the value is:
[tex]\[ \tan 13^\circ \approx 0.23086819112556312 \][/tex]
### c) [tex]\(\sin 257^\circ\)[/tex]
Using the sine's periodic property and reference angle:
[tex]\[ \sin (360^\circ - \theta) = -\sin \theta \][/tex]
For [tex]\(\theta = 257^\circ\)[/tex]:
[tex]\[ 257^\circ = 360^\circ - 103^\circ \][/tex]
[tex]\[ \sin 257^\circ = -\sin 103^\circ \][/tex]
And,
[tex]\[ \sin 103^\circ = \cos (90^\circ - 103^\circ) = \cos 13^\circ \][/tex]
So,
[tex]\[ \sin 257^\circ = -\cos 13^\circ = -b \][/tex]
Thus, the value is:
[tex]\[ \sin 257^\circ \approx -0.9743700647852351 \][/tex]
### d) [tex]\(\cos 347^\circ\)[/tex]
Using the cosine's periodic property and reference angle:
[tex]\[ \cos (360^\circ - \theta) = \cos \theta \][/tex]
For [tex]\(\theta = 347^\circ\)[/tex]:
[tex]\[ 347^\circ = 360^\circ - 13^\circ \][/tex]
[tex]\[ \cos 347^\circ = \cos 13^\circ \][/tex]
Given that [tex]\(\cos 13^\circ = b\)[/tex], we have:
[tex]\[ \cos 347^\circ = b \][/tex]
Thus, the value is:
[tex]\[ \cos 347^\circ \approx 0.9743700647852351 \][/tex]
