Given the right triangle below, evaluate tanθ\tan\thetatanθ.

tan ⁡ θ = 3 4 \tan\theta=\frac34 tan θ = 4 3

Given the right triangle below, evaluate tan θ \tan\theta tan θ .

everyone. So let's try solving this problem. Here it says, given the right triangle below, evaluate the tangent of theta, and we've got our angle theta right there. Now, to solve this problem, we need to recall SOHCAHTOA. And SOHCAHTOA is this memory tool that reminds us of the main trig functions that you're going to see, which is sine, cosine, and tangent. Now in this problem, we are dealing with the tangent of theta, so what I'm going to do is focus on this relationship right here. This is that tangent is opposite over adjacent. So we can write that our tangent the the tangent of our angle theta, this angle, is equal to the opposite side of the triangle divided by the adjacent side of the triangle. Now looking at this angle, the opposite side is going to be the side that opposes the angle, which is 12. And the adjacent side is going to be this side, which is right next to the angle, which is 16. Also, something to remember is that even though this side of the triangle is also close to the angle, the longest side of the triangle is always called the hypotenuse. So we know the adjacent side is going to be shorter than the hypotenuse, so that's how we knew this was the adjacent side. Now we have that our tangent of theta is equal to 12 over 16, but it looks like this fraction can actually reduce because 12 is the same thing as three times four. Right? And then this is all equal to our tangent of theta. And then 16 is the same thing as four times four. Notice how we can actually cancel a four in this equation, meaning that our tangent of theta is going to equal three fourths because we can see that that's what things reduce to when we cancel the fours. So as you can see this right here is going to be the simplified relationship for the tangent of theta and that is the answer to this problem. So hope you found this video helpful, thanks for watching.

Given the right triangle below, evaluate tan ⁡ θ \tan\theta tan θ .

tan ⁡ θ = 3 4 \tan\theta=\frac34 tan θ = 4 3

tan ⁡ θ = 3 5 \tan\theta=\frac35 tan θ = 5 3

tan ⁡ θ = 4 5 \tan\theta=\frac45 tan θ = 5 4

tan ⁡ θ = 4 3 \tan\theta=\frac43 tan θ = 3 4

12. Trigonometric Functions

Trigonometric Functions on Right Triangles

Business Calculus

12. Trigonometric Functions

Trigonometric Functions on Right Triangles

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Multiple Choice

Given the right triangle below, evaluate $\tan\theta$ .

$\tan\theta=\frac35$

$\tan\theta=\frac45$

$\tan\theta=\frac43$

$\tan\theta=\frac34$

Verified step by step guidance

Step 1: Recall the definition of tangent in a right triangle. The tangent of an angle θ is defined as the ratio of the length of the opposite side to the length of the adjacent side: tan(θ) = opposite / adjacent.

Step 2: Identify the sides of the triangle relative to the angle θ. From the image, the side opposite θ is 12, and the side adjacent to θ is 16.

Step 3: Substitute the values of the opposite and adjacent sides into the formula for tangent: tan(θ) = 12 / 16.

Step 4: Simplify the fraction 12 / 16 by dividing both the numerator and denominator by their greatest common divisor, which is 4. This simplifies to tan(θ) = 3 / 4.

Step 5: Verify that the simplified ratio matches one of the given options. The correct answer is tan(θ) = 3 / 4.

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