Tangent - ACT Math

To solve this question, you must know SOHCAHTOA. This acronym can be broken into three parts to solve for the sine, cosine, and tangent.

We can use this information to solve our identity.

Dividing by a fraction is the same as multiplying by its reciprocal.

The sine divided by cosine is the tangent of the angle.

The question asks for you to find the perimeter of the given figure. The figure has twelve sides total, of two varying lengths. One length is given to you, 4. The other length must be solved for using either the sine or tangent functions. However, one can arrive to answer more quickly by recognizing that the drawn triangle is actually a 3-4-5 triangle, where 3, 4, and 5 corresponds to each of the sides of the triangle. This is a pythagorean triple and this ratio should be easily remembered.

Thus if 3 is the missing side, and there are eight sides of length 3 and four sides of length 4, one can arrive to the answer:

You can draw out your scenario like a triangle:

Now, you know that this means:

Using your calculator, you can utilize the inverse function to calculate the degree measure of the angle:

This rounds to degrees.

Tangent = Opposite / Adjacent

The tangent of an angle x is defined as

Substituting the given values for cos x and sin x, we get

Use the definition of the tangent and plug in the values given:

tangent C = Opposite / Adjacent = AB / BC = 3 / 7

Therefore, BC = 7.

The cosine of the angle is and since that is a reduced fraction, we know the hypotenuse is and the adjacent side equals .

The sine of the angle equals , and since the hyptenuse is already we know that we must multiply the numerator and denominator by to get the common denominator of . Therefore, the opposite side equals .

Since , the answer is .

The function is related to . The range of the tangent function is .

The range of is unaffected by the amplitude and the y-intercept. Therefore, the answer is .

For tangent and cotangent functions, the range is always all real numbers.

The range of a function is every value that the funciton's results take. For tangent and cotangent, the function spans from and so the range is:

Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the -axis as your reference point for your angle. (Hence, it is called the "reference angle.")

Now, it is easiest to think of this like you are drawing a little triangle in the fourth quadrant of the Cartesian plane. It would look like:

So, the tangent of an angle is:

or, for your data, or . However, since is in the fourth quadrant, your value must be negative. (The tangent function is negative in that quadrant.) This makes the correct answer .

Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the -axis as your reference point for your angle. (Hence, it is called the "reference angle.")

Now, it is easiest to think of this like you are drawing a little triangle in the second quadrant of the Cartesian plane. It would look like:

So, the tangent of an angle is:

or, for your data, .

This is . Rounding, this is . However, since is in the second quadrant, your value must be negative. (The tangent function is negative in that quadrant.) Therefore, the answer is .

You can begin by imagining a little triangle in the second quadrant to find your reference angle. It would look like this:

The tangent of an angle is:

For our data, this is:

Now, since this is in the second quadrant, the value is negative, given the periodic nature of the tangent function.

Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the -axis as your reference point for your angle. (Hence, it is called the "reference angle.")

Now, it is easiest to think of this like you are drawing a little triangle in the fourth quadrant of the Cartesian plane. It would look like:

So, the tangent of an angle is:

or, for your data, or . However, since is in the fourth quadrant, your value must be negative. (The tangent function is negative in that quadrant.) This makes the correct answer .

Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the -axis as your reference point for your angle. (Hence, it is called the "reference angle.")

Now, it is easiest to think of this like you are drawing a little triangle in the second quadrant of the Cartesian plane. It would look like:

So, the tangent of an angle is:

or, for your data, .

This is . Rounding, this is . However, since is in the second quadrant, your value must be negative. (The tangent function is negative in that quadrant.) Therefore, the answer is .

You can begin by imagining a little triangle in the second quadrant to find your reference angle. It would look like this:

The tangent of an angle is:

For our data, this is:

Now, since this is in the second quadrant, the value is negative, given the periodic nature of the tangent function.

With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the opposite and adjacent sides of the triangle with relation to the angle. With this information, we can use the tangent function to find the angle.

With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the opposite and adjacent sides of the triangle with relation to the angle. With this information, we can use the tangent function to find the angle.

With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the opposite and adjacent sides of the triangle with relation to the angle. With this information, we can use the tangent function to find the angle.

You can draw your scenario using the following right triangle:

Recall that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side of the triangle. You can solve for the angle by using an inverse tangent function:

or .