find tan 30 degrees and tan 45 degrees using the 45-45-90 triangle therom and
30-60-90 triangle theorm
In an easily accessible part of your notebook
make a sketch of a 30-60-90 triangle, and put in the values
1 opposite the 30°
√3 opposite the 60°
2 opposite the hypotenuse
MEMORIZE THESE VALUES
(the smallest is opposite the smallest angle, etc..)
Do the same for the 45-45-90 using sides
1 - 1 - √2
now tanØ = opposite/adjacent or y/x
tan 30 = 1/√3 , look at your triangle
tan 45 = 1/1 = 1
check with your calculator
To find the values of tan 30 degrees and tan 45 degrees using the 45-45-90 and 30-60-90 triangle theorems, follow these steps:
1. Recall the properties of a 45-45-90 triangle:
- The two legs of a 45-45-90 triangle are congruent.
- The hypotenuse is the length of one leg multiplied by the square root of 2.
2. Start by drawing a right triangle and label the angles and sides accordingly.
3. Determine the ratio of the tangent function for 30 degrees in the 30-60-90 triangle:
- In a 30-60-90 triangle, the ratio of the sides opposite and adjacent to the 30 degrees angle is 1:√3:2.
- The tangent function (tan) is defined as the ratio of the side opposite the angle to the side adjacent to the angle.
So, in a 30-60-90 triangle, tan 30 degrees is equal to the opposite side divided by the adjacent side.
tan 30 degrees = (1/2)/(√3/2) = 1/√3 = √3/3
4. Determine the ratio of the tangent function for 45 degrees in the 45-45-90 triangle:
- In a 45-45-90 triangle, the ratio of the sides opposite and adjacent to the 45 degrees angle is 1:1:√2.
- The tangent function (tan) is defined as the ratio of the side opposite the angle to the side adjacent to the angle.
So, in a 45-45-90 triangle, tan 45 degrees is equal to the opposite side divided by the adjacent side.
tan 45 degrees = 1/1 = 1
Therefore, using the 45-45-90 triangle theorem and the 30-60-90 triangle theorem:
tan 30 degrees = √3/3
tan 45 degrees = 1
To find the tangent of 30 degrees and 45 degrees using the 45-45-90 triangle theorem and the 30-60-90 triangle theorem, we can follow these steps:
1. Recall the 45-45-90 triangle theorem: In a 45-45-90 triangle, the two legs (the sides opposite the 45-degree angles) are congruent, and the hypotenuse (the side opposite the 90-degree angle) is equal to √2 times the length of the legs.
2. Let's start with finding the tangent of 45 degrees. In a 45-45-90 triangle, the tangent of one of the 45-degree angles is defined as the ratio of the length of the opposite leg to the length of the adjacent leg.
Since the two legs of a 45-45-90 triangle are congruent, let's assign a variable, "x", to represent the length of each leg. Therefore, the opposite leg is "x" and the adjacent leg is also "x".
Using the tangent formula, the tangent of 45 degrees is:
tan(45°) = opposite / adjacent
= x / x
= 1
So, tan(45°) = 1.
3. Now, let's find the tangent of 30 degrees using the 30-60-90 triangle theorem. In a 30-60-90 triangle, the length of the shorter leg (opposite the 30-degree angle) is half the length of the hypotenuse, and the length of the longer leg (opposite the 60-degree angle) is √3 times the length of the shorter leg.
Let's assign a variable, "y", to represent the length of the shorter leg, which is opposite the 30-degree angle. Therefore, the hypotenuse is "2y" and the longer leg is "√3y".
Using the tangent formula, the tangent of 30 degrees is:
tan(30°) = opposite / adjacent
= y / (√3y)
= 1 / √3
To simplify the expression, we can rationalize the denominator by multiplying both the numerator and denominator by √3:
tan(30°) = 1 / √3 * √3 / √3
= √3 / 3
So, tan(30°) = √3 / 3.
In summary,
- tan(45°) = 1 (using the 45-45-90 triangle theorem)
- tan(30°) = √3 / 3 (using the 30-60-90 triangle theorem)