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In this article, you would understand what happens when trigonometric identities are either doubled or halved.
Trigonometric functions can be doubled but not in the same way as normal numbers are doubled.
If you have the expression 3y and you are to double it, it is easy to multiply 3y by 2 to get 6y. Note that sin30° is 0.5 and doubling the angle gives 60°, but sin60° would not give you 1. As normal mathematical operations would multiply 0.5 by 2 to give 1, trigonometric identities would require their own formula to double their function.
We intend on finding the formula for sin2θ. Note that
Recall that,
Now take we get
The double angle formula for the sine function is given by
Findusing the double angle formula.
Solution:
We have
Thus
but,
Then,
Given that, find
if
Solution:
We have sinθ in the given, but in order to apply our formula, we need to find cosθ.
Recall that,
Thus,
We take the square root of both sides to get,
Note that the range of the angle is between 90° and 180°, this means that θ is in the second quadrant. The cosine of angles in the second quadrant has negative values. Thus,
Now we have to apply our double angle formula,
We will now develop a formula for a double-angle for the cosine function. We derive three equal formulas.
We first note that
Now, recall that
By takingwe get
Thus, we derive the first formula for ,
We now recall the identity
Now we replace this with the obtained formula forto get
Thus, the second formula for is
In a similar way, we have .
Substituting the value of into the formula of
, we have
Thus, the third formula for is
The double angle formulas for the cosine function are given by,
Given that , find
if
Solution:
Method 1.
The direct way to find is to use the formula
, since we are given the value of
.
So,
Method 2.
We can use either of the other formulas to find , we will use
We need thus to find
.
We recall that , thus
Taking the square root of both sides, we get
Note that , this means that
is in the second quadrant. The cosine of angles in the second quadrant have negative values. Thus,
So, we can apply our formula
For , find
when
Solution:
In solving this problem, it is faster to use the formula.
Thus,
We will develop a formula for a double-angle of the tangent function.
We recall that
and,
Thus,
Substituting and
by their expressions, we get
To simplify this further, we divide both the numerator and the denominator of the right-hand side of the equation by , to get
The double angle formula for the tangent function is given by,
Given that find
if
Solution:
We need to find, this means that we firstly have to find
.
We recall that , thus
. Replacing
by its value, we get
Note that , this means that
is in the second quadrant. The cosine of angles in the second quadrant have negative values. Thus,
.
Therefore,
Thus,
Secant, cosecant, cotangent functions are the reciprocals of cosine, sine and tangent respectively. In order to derive their double-angle formulas, you just need to find the multiplicative inverse of the corresponding double-angle formulas.
We recall by the definition of the secant function that
so
but from the double angle formula for cosine we have , thus
Now let us express in terms of
and
.
In fact, and
, thus we have
The double angle formula for the secant function is given by,
We recall by the definition of the secant function that
so
but from the double angle formula for sine function, we have, thus
The double angle formula for the cosecant function is given by,
We recall by the definition of the secant function that
so
We recall from the double angle formula for the tangent function that , we have
The double angle formula for the cotangent function is given by,
Given that find
and
given that
Solution:
We have the value of sinθ, but in order to apply these formulas, we need to find cosθ.
We recall that,
Thus, but since
thus
.
So
Hence, we have
Trigonometric functions can be halved but not in the same manner normal numbers are halved. If you have the expression 6y and you are to half it, it is easy to multiply 6y by 0.5 to get 3y. Note that sin30° is 0.5 and halving the angle gives 15 degrees, but sin15° would not give you 0.25. As normal mathematical operations would multiply 0.5 by 0.5 (half) to give 0.25, trigonometric identities would require their own formula to half its function.
To find , we recall first that
Let , thus
In order to isolate , subtract 1 from both sides, to get
Taking the square root of both sides of the equation, we get
The half-angle formula for the sine function is given by,
If , and 90°<θ<180°, find
.
Solution:
Since , thus
Hence,
Thus, to find, we need to find cosθ. Recall that
Since
Then,
Now we can substitute the value of cosθ into our equation
Recall that
Where,
Therefore
Divide both sides by 2
Find the square root of both sides of the equation
Thus
Given that
for , find
.
Solution:
To begin, get the value of cosθ.
Note that
Thus
Recall from the question that θ falls within the third quadrant, hence cosine values would be negative. Thus
Note before
So by substituting the value of cosθ we get
Multiply the right hand side of the equation by (rationalization of surds)
Now θ has been halved, the conditions would change too by
Angles here fall in the third quadrant.
Dividing that by 2 you have
falls in the second quadrant and cosθ is negative in the second quadrant.
Knowing that
Then
Multiply the right-hand side of the equation by and you would have
Recall that
Then
Find when
.
Solution:
With the value given, the opposite and adjacent are 4 and 3 respectively. Using the Pythagoras theorem we shall arrive at a value for the hypotenuse.
Now we have the value of the hypotenuse, then
You can now apply the formula
As mentioned earlier, secant, cosecant, cotangent are the inverse of cosine, sine and tangent respectively. So as to derive their half-angle formulas, you just need to find the multiplicative inverse of the corresponding half-angle formulas. Thus the half-angle formula of secant becomes:
the half-angle formula of cosecant becomes:
and the half-angle formula of cotangent becomes:
This is the same as
If , find values for
,
and
.
Solution:
Since,
and
Then,
Knowing that;
Therefore;
Since the values for cosθ as well as sinθ have been found, it is easier to find the half angles of sec, cosec and cot. Thus half-angle of sec becomes:
For half-angle of cosec
And for the half-angle of cot
Here are a few examples that show the application of double-angle and half-angle formulas.
Solve the for θ in
Solution:
Recall that
Substitute into the equation. Therefore,
This means that
or
Now in finding θ, we have to find both arcsinθ and arccosθ. Therefore,
However, -2 goes beyond the possible values for arccosθ. Hence,
is invalid
Thus the value of θ is 270°.
If
find
Solution:
The first thing to do is to find cosϕ. Knowing that
Now, substitute the value of sinϕ to find cosϕ.
Recall that
Hence,
Double-angle and half-angle formulas are formulas used in finding the trigonometric values for angles that are doubled or halved.
Double-angle and half-angle formulas are used by applying directly the formulas when finding double/half-angle trigonometric identities.
An example of a double angle formula is sin2A = 2sinAcosA, while that of half-angle is sinA/2 = square root ((1-cosA)/2).
To derive double angle formulas you would need to apply the sum of trigonometric function formulas.
The types of double angle formulas are those of sin, cos, sec, cosec, tan and cot.
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