ACT Math : Trigonometry
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Example Questions
Example Question #1 : How To Find The Sine Of A Missing Side
In the triangle below, 
Because 
Using the Pythagorean theorem,
Example Question #2 : How To Find The Sine Of A Missing Side
What is the value of 
We know that the sine of an angle is:
Therefore, we can write for this question:
This allows us to solve for 
Rounding, this gives us 
Example Question #3 : How To Find The Sine Of A Missing Side
A right triangle has leg lengths 
Using SOHCAHTOA, the sine of an angle is simply the length of the side opposite to it over the hypotenuse; however, we do not have the length of the hypotenuse yet.
Using the Pythagorean Theorem, we can solve for it:
So, the sine of this angle is:
Example Question #1 : How To Find An Angle With Sine
Simplify: (sinΘ + cosΘ)2
cos2Θ -1
1 + sin2Θ
None of the answers are correct
2sinΘcosΘ -1
1 + cos2Θ
1 + sin2Θ
Using the foil method, multiply. Simplify using the Pythagorean identity sin2Θ + cos2Θ = 1 and the double angle identity sin2Θ = 2sinΘcosΘ.
Example Question #2 : How To Find An Angle With Sine
For the triangle 
, find 
Note: The triangle is not necessarily to scale
None of the other answers
To solve this equation, it is best to remember the mnemonic SOHCAHTOA which translates to Sin = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, and Tangent = Opposite / Adjacent. Looking at the problem statement, we are given the side opposite of the angle we are trying to find as well as the hypotenuse. Therefore, we will be using the SOH part of our mnemonic. Inserting our values, this becomes 
Example Question #3 : How To Find An Angle With Sine
A fifteen foot ladder is leaned up against a twelve foot building, reaching the top of the building. What is the angle made between the ladder and the ground? Round to the nearest hundredth of a degree.
You can draw your scenario using the following right triangle:
Recall that the sine of an angle is equal to the ratio of the opposite side to the hypotenuse of the triangle. You can solve for the angle by using an inverse sine function:
Example Question #1 : How To Find An Angle With Sine
What is the value of 
Recall that the sine of an angle is equal to the ratio of the opposite side to the hypotenuse of the triangle. You can solve for the angle by using an inverse sine function:
Example Question #1 : Cosine
In the above triangle, 
With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the adjacent and hypotenuse sides of the triangle with relation to the angle. With this information, we can use the cosine function to find the angle.
Example Question #2 : Cosine
For the above triangle, 
With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the adjacent and hypotenuse sides of the triangle with relation to the angle. With this information, we can use the cosine function to find the angle.
Example Question #1 : How To Find An Angle With Cosine
For the above triangle, 
This triangle cannot exist.
This triangle cannot exist.
With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the adjacent and hypotenuse sides of the triangle with relation to the angle. However, if we plug the given values into the formula for cosine, we get:
This problem does not have a solution. The sides of a right triangle must be shorter than the hypotenuse. A triangle with a side longer than the hypotenuse cannot exist. Similarly, the domain of the arccos function is 
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