Find the best answers to your questions with the help of IDNLearn.com's expert contributors. Ask your questions and get detailed, reliable answers from our community of experienced experts.
Sagot :
To find the exact values of [tex]\(\sin \theta\)[/tex], [tex]\(\sec \theta\)[/tex], and [tex]\(\tan \theta\)[/tex] when given the point [tex]\((-5, -3)\)[/tex] on the terminal side of [tex]\(\theta\)[/tex], follow these steps:
1. Calculate the hypotenuse [tex]\( r \)[/tex]:
The hypotenuse [tex]\( r \)[/tex] is found using the Pythagorean theorem, where [tex]\( r \)[/tex] is the distance from the origin to the point [tex]\((-5, -3)\)[/tex].
[tex]\[ r = \sqrt{x^2 + y^2} \][/tex]
Substituting [tex]\( x = -5 \)[/tex] and [tex]\( y = -3 \)[/tex]:
[tex]\[ r = \sqrt{(-5)^2 + (-3)^2} \][/tex]
[tex]\[ r = \sqrt{25 + 9} \][/tex]
[tex]\[ r = \sqrt{34} \][/tex]
Hence,
[tex]\[ r \approx 5.830951894845301 \][/tex]
2. Determine [tex]\(\sin \theta\)[/tex]:
By definition, [tex]\(\sin \theta\)[/tex] is the ratio of the opposite side over the hypotenuse.
[tex]\[ \sin \theta = \frac{y}{r} \][/tex]
Substituting [tex]\( y = -3 \)[/tex] and [tex]\( r \approx 5.830951894845301 \)[/tex]:
[tex]\[ \sin \theta = \frac{-3}{5.830951894845301} \][/tex]
[tex]\[ \sin \theta \approx -0.5144957554275265 \][/tex]
3. Determine [tex]\(\sec \theta\)[/tex]:
By definition, [tex]\(\sec \theta\)[/tex] is the reciprocal of [tex]\(\cos \theta\)[/tex], where [tex]\(\cos \theta\)[/tex] is the ratio of the adjacent side over the hypotenuse.
[tex]\[ \sec \theta = \frac{r}{x} \][/tex]
Substituting [tex]\( r \approx 5.830951894845301 \)[/tex] and [tex]\( x = -5 \)[/tex]:
[tex]\[ \sec \theta = \frac{5.830951894845301}{-5} \][/tex]
[tex]\[ \sec \theta \approx -1.1661903789690602 \][/tex]
4. Determine [tex]\(\tan \theta\)[/tex]:
By definition, [tex]\(\tan \theta\)[/tex] is the ratio of the opposite side over the adjacent side.
[tex]\[ \tan \theta = \frac{y}{x} \][/tex]
Substituting [tex]\( y = -3 \)[/tex] and [tex]\( x = -5 \)[/tex]:
[tex]\[ \tan \theta = \frac{-3}{-5} \][/tex]
[tex]\[ \tan \theta = 0.6 \][/tex]
Therefore, the values are:
[tex]\[ \sin \theta \approx -0.5144957554275265 \][/tex]
[tex]\[ \sec \theta \approx -1.1661903789690602 \][/tex]
[tex]\[ \tan \theta \approx 0.6 \][/tex]
1. Calculate the hypotenuse [tex]\( r \)[/tex]:
The hypotenuse [tex]\( r \)[/tex] is found using the Pythagorean theorem, where [tex]\( r \)[/tex] is the distance from the origin to the point [tex]\((-5, -3)\)[/tex].
[tex]\[ r = \sqrt{x^2 + y^2} \][/tex]
Substituting [tex]\( x = -5 \)[/tex] and [tex]\( y = -3 \)[/tex]:
[tex]\[ r = \sqrt{(-5)^2 + (-3)^2} \][/tex]
[tex]\[ r = \sqrt{25 + 9} \][/tex]
[tex]\[ r = \sqrt{34} \][/tex]
Hence,
[tex]\[ r \approx 5.830951894845301 \][/tex]
2. Determine [tex]\(\sin \theta\)[/tex]:
By definition, [tex]\(\sin \theta\)[/tex] is the ratio of the opposite side over the hypotenuse.
[tex]\[ \sin \theta = \frac{y}{r} \][/tex]
Substituting [tex]\( y = -3 \)[/tex] and [tex]\( r \approx 5.830951894845301 \)[/tex]:
[tex]\[ \sin \theta = \frac{-3}{5.830951894845301} \][/tex]
[tex]\[ \sin \theta \approx -0.5144957554275265 \][/tex]
3. Determine [tex]\(\sec \theta\)[/tex]:
By definition, [tex]\(\sec \theta\)[/tex] is the reciprocal of [tex]\(\cos \theta\)[/tex], where [tex]\(\cos \theta\)[/tex] is the ratio of the adjacent side over the hypotenuse.
[tex]\[ \sec \theta = \frac{r}{x} \][/tex]
Substituting [tex]\( r \approx 5.830951894845301 \)[/tex] and [tex]\( x = -5 \)[/tex]:
[tex]\[ \sec \theta = \frac{5.830951894845301}{-5} \][/tex]
[tex]\[ \sec \theta \approx -1.1661903789690602 \][/tex]
4. Determine [tex]\(\tan \theta\)[/tex]:
By definition, [tex]\(\tan \theta\)[/tex] is the ratio of the opposite side over the adjacent side.
[tex]\[ \tan \theta = \frac{y}{x} \][/tex]
Substituting [tex]\( y = -3 \)[/tex] and [tex]\( x = -5 \)[/tex]:
[tex]\[ \tan \theta = \frac{-3}{-5} \][/tex]
[tex]\[ \tan \theta = 0.6 \][/tex]
Therefore, the values are:
[tex]\[ \sin \theta \approx -0.5144957554275265 \][/tex]
[tex]\[ \sec \theta \approx -1.1661903789690602 \][/tex]
[tex]\[ \tan \theta \approx 0.6 \][/tex]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.
