Right triangles with certain acute angle measures are considered noteworthy. These include the following angle relationships.
Although the first and third relationships contain the same angles, they are considered different because the reference angle in each is different — 30∘ in the first and 60∘ in the third. The sine, cosine, and tangents of these triangles can be useful when finding unknown side lengths.
Angle θ | 30∘ | 45∘ | 60∘ |
---|---|---|---|
sin(θ) | 21 | 22 | 23 |
cos(θ) | 23 | 22 | 21 |
tan(θ) | 33 | 1 | 3 |
These special measures are justified below.
Consider the 30∘ - 60∘ - 90∘ triangle. Suppose an equilateral triangle has side lengths of 1.
Bisecting the apex angle yields the following 30∘ - 60∘ - 90∘ triangle.
The value of h — the length of the third side — can be found using the Pythagorean Theorem. Here, a=21,b=h, and c=1.
Since, h=23, the 30∘ - 60∘ - 90∘ triangle can be redrawn as follows.
Using θ=30∘, the following relationships can be concluded.
sin(30∘)=21/1 =21cos(30∘)=23/1 =23tan(30∘)=21/23=33
Using the same triangle, the values for a 60∘ - 30∘ - 90∘ triangle can be determined.
sin(60∘)=23/1 = 23cos(60∘)=21/1 =21tan(60∘)=23/21=3
Suppose an isosceles triangle has a hypotenuse of 1 and base angles that measure 45∘.
Because the triangle is isosceles, its legs have equal measure. Because they are unknown, x can be used to represent them. The Pythagorean Theorem can be used to determine the value of x.
The 45∘ - 45∘ - 45∘ triangle can be redrawn as follows.
Using θ=45∘, the following relationships can be concluded.
sin(45∘)= 22/1 =22cos(45∘)= 22/1 =22tan(45∘)=22/22= 1
As was shown above, because the hypotenuse of these special right triangles is 1,
sin(θ)=opp and cos(θ)=adj,for θ=30∘,45∘,60∘.
Therefore, these values can be used to determine unknown side lengths.By placing right triangles with one of their legs along the x-axis, the coordinates of the vertex not on the axis can be found using the legs of the triangle.
In the first quadrant, the coordinates are the lengths of the triangle's legs. If the length of the hypotenuse is 1 the legs of the special right triangles are given directly by the sine and cosine value of the angle at the origin and the coordinates are also the sine and cosine values.
Since the hypotenuse of these triangles is 1, the marked points all lie on a circle with radius 1 and a center at the origin.
This is called the unit circle. The coordinates of any point on this circle are the cosine and sine values of the angle the segment from the point to the origin creates with the positive x-axis.
x=cos(θ) and y=sin(θ)
A radian is, like a degree, an angle unit. If the arc length of a circle sector is equal to the radius of the circle, the angle created is 1 radian (rad), which corresponds to roughly 57.3∘.
If the arc length is 2 radii, the angle is 2 radians, etc. Thus, radians describe the number of radii an angle creates on a circle.
Use special right triangles and the unit circle to find the following trigonometric values. sin(45∘)tan(π)cos(150∘)
Let's find the trigonometric values one at a time.
The value sin(45∘) is the y-coordinate of the point on the unit circle that, together with the positive x-axis, creates the angle 45∘.
Here, we have a right triangle with the angles 45∘ and 45∘. The hypotenuse is 1, so we know that the legs of the triangle are both 22. That means the point is (22,22).
Therefore, sin(45∘) is 22.
Notice there is no degree symbol. This means the angle is given in radians. π radians is half a revolution, meaning 180∘.
Let's draw the angle 150∘ from the positive x-axis and mark the point it corresponds to on the unit circle.
An angle of 180∘ would span from the positive x-axis to the negative. Hence, a 30∘ special right triangle can be drawn from the negative x-axis to the terminal side of the 150∘ angle. Recall that this triangle has legs that measure 21 and 23.
Since cos(150∘) is the x-coordinate of the point, cos(150∘)=-23.
Since both degrees and radians are used to measure angles, it's useful to be able to translate between them.
The arc length for one revolution around a circle is also known as the circumference of the circle, 2πr. By dividing this with the length of the arc corresponding to one radian, r, the number of radians for one revolution is obtained. r2πr=2π Thus, one revolution is 2π radians. In degrees, this is 360∘, leading to the relation 360∘=2π rad. Dividing both sides by 2 gives 180∘=π rad. Thus, 180∘ correspond to π radians.
From the relation 180∘=π rad it's possible to find two rules, by dividing both sides by either 180 or π.
To get an expression for 1 rad, divide both sides by π.
1 radian corresponds to π180≈57.3∘.
Convert 45∘ to radians and 2π rad to degrees.
To convert an angle from degrees to radians we can use 1∘=180π rad. 45∘ should be multiplied with 180π rad.
Thus, 45∘ is equal to 4π radians. We could use the caluclator to get a numerical value here, but there are infinitely many decimals, so we'll keep it in its exact form: 45∘=4π rad. To convert an angle from radians to degrees we can use that 1 rad=π180∘. Therefore, if we multiply 2π with π180∘, we'll get how many degrees it corresponds to.