All ACT Math Resources
Example Questions
Example Question #21 : Tangent
For the right triangle shown below, what is the value of
?
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.
Example Question #22 : Tangent
For triangle , what is the cotangent of angle ?
The cotangent of the angle of a triangle is the adjacent side over the opposite side. The answer is
Example Question #22 : Trigonometry
What is the tangent of the angle formed between the origin and the point if that angle is formed with one side of the angle beginning on the -axis and then rotating counter-clockwise to ? Round to the nearest hundredth.
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 third 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 . Since is in the third quadrant, your value must be positive, as the tangent function is positive in that quadrant.
Example Question #3 : How To Find Positive Tangent
What is the tangent of the angle formed between the origin and the point if that angle is formed with one side of the angle beginning on the -axis and then rotating counter-clockwise to ? Round to the nearest hundredth.
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 third quadrant of the Cartesian plane. It would look like:
So, the tangent of an angle is:
or, for your data, , or . Since is in the third quadrant, your value must be positive, as the tangent function is positive in this quadrant.
Example Question #2 : How To Find Positive Tangent
A ramp is being built at an angle of from the ground. It must cover horizontal feet. What is the length of the ramp? Round to the nearest hundredth of a foot.
Based on our information, we can draw this little triangle:
Since we know that the tangent of an angle is , we can say:
This can be solved using your calculator:
or
Now, to solve for , use the Pythagorean Theorem, , where and are the legs of a triangle and is the triangle's hypotenuse. Here, , so we can substitute that in and rearrange the equation to solve for :
Substituting in the known values:
, or approximately . Rounding, this is .
Example Question #1 : How To Find Negative Tangent
What is the tangent of the angle formed between the origin and the point if that angle is formed with one side of the angle beginning on the -axis and then rotating counter-clockwise to ?
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.
Example Question #2 : How To Find Negative Tangent
What is the tangent of the angle formed between the origin and the point if that angle is formed with one side of the angle beginning on the -axis and then rotating counter-clockwise to ? Round to the nearest hundredth.
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 .
Example Question #3 : How To Find Negative Tangent
What is the tangent of the angle formed between the origin and the point if that angle is formed with one side of the angle beginning on the -axis and then rotating counter-clockwise to ? Round to the nearest hundredth.
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 .
Example Question #1 : How To Find The Period Of The Tangent
Which of the following equations represents a tangent function with a period that is radians?
The standard period of a tangent function is radians. In other words, it completes its entire cycle of values in that many radians. To alter the period of the function, you need to alter the value of the parameter of the trigonometric function. You multiply the parameter by the number of periods that would complete in radians. With a period of , you are quadrupling your method. Therefore, you will have a function of the form:
Since and do not alter the period, these can be anything.
Therefore, among your options, is correct.
Example Question #1 : How To Find The Period Of The Tangent
Which of the following represents a tangent function that has a period half that of one with a period of ?
The standard period of a tangent function is radians. In other words, it completes its entire cycle of values in that many radians. To alter the period of the function, you need to alter the value of the parameter of the trigonometric function. You multiply the parameter by the number of periods that would complete in radians. With a period of , you are multiplying your parameter by . Now, half of this would be a period of . Thus, you will have a function of the form:
Since and do not alter the period, these can be anything.
Therefore, among your options, is correct.
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