10. Rewrite 2tan 23° / 1 - tan223° as a single function of an angle.
I'd appreciate it if someone could help me out with this question! Don't just give me the answer though, please and thank you!
Well, well, well, if it isn't a math problem in need of a clown's touch! Don't worry, my friend, Clown Bot is here to lend a hand!
Let's break it down step by step, shall we?
Step 1: Let's use the identity tan(θ) = sin(θ) / cos(θ) to rewrite the expression:
2tan(23°) / (1 - tan²(23°))
Step 2: Substitute sin(θ) / cos(θ) for tan(θ):
2(sin(23°) / cos(23°)) / (1 - (sin²(23°) / cos²(23°)))
Step 3: To simplify this further, let's multiply through by cos²(23°):
2(sin(23°) * cos(23°)) / (cos²(23°) - sin²(23°))
Step 4: Now, remember the trig identity cos²(θ) - sin²(θ) = cos(2θ). Let's apply that:
2(sin(23°) * cos(23°)) / cos(46°)
Et voilà! We have now rewritten 2tan(23°) / (1 - tan²(23°)) as a single function of an angle: 2(sin(23°) * cos(23°)) / cos(46°).
Remember, sometimes math problems can be a little tricky, but with a touch of humor and a little bit of clown magic, you can conquer them!
To rewrite 2tan 23° / 1 - tan^2 23° as a single function of an angle, we can use the trigonometric identity:
tan^2θ + 1 = sec^2θ
Rearranging the equation, we have:
1 - tan^2θ = sec^2θ - tan^2θ
Now, substituting 23° for θ, we get:
1 - tan^2 23° = sec^2 23° - tan^2 23°
Using this identity, we can rewrite the expression as a single function of an angle:
2tan 23° / (1 - tan^2 23°) = 2tan 23° / (sec^2 23° - tan^2 23°)
To rewrite the expression 2tan 23° / 1 - tan^2 23° as a single function of an angle, we can use the identity tan^2 θ + 1 = sec^2 θ. Here's how we can do it step-by-step:
Step 1: Start with the given expression.
2tan 23° / 1 - tan^2 23°
Step 2: Recall the identity tan^2 θ + 1 = sec^2 θ.
Rewrite the expression using this identity:
2tan 23° / sec^2 23° - tan^2 23°
Step 3: Since sec^2 θ = 1 / cos^2 θ, let's replace sec^2 23° with 1 / cos^2 23°.
2tan 23° / (1 / cos^2 23°) - tan^2 23°
Step 4: Simplify the expression by multiplying the numerator and denominator by cos^2 23°.
[2tan 23° * cos^2 23°] / 1 - tan^2 23°
Step 5: Use the identity tan θ = sin θ / cos θ.
Replace tan 23° with sin 23° / cos 23°:
[2(sin 23° / cos 23°) * cos^2 23°] / 1 - (sin 23° / cos 23°)^2
Step 6: Simplify the expression by multiplying and canceling out terms.
[2(sin 23° * cos^2 23°)] / cos^2 23° - sin^2 23°
Step 7: Expand and simplify the expression.
2 sin 23° * cos^2 23° / cos^2 23° - sin^2 23°
Step 8: Simplify the denominator of the expression by factoring it.
2 sin 23° * cos^2 23° / (cos 23° + sin 23°)(cos 23° - sin 23°)
Step 9: Cancel out the common factor of cos 23° from the numerator and denominator.
2 sin 23° cos 23° / (cos 23° + sin 23°)
Finally, we have rewritten the expression 2tan 23° / 1 - tan^2 23° as a single function of an angle, which is 2 sin 23° cos 23° / (cos 23° + sin 23°).
If you fix your typo:
2tan 23° / (1 - tan^2 23°)
and recall your double-angle formula for tan(x) you will see what you need to do.