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The trigonometric ratios cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent, respectively.
csc θ=1/sin θ
sec θ=1/cos θ
cot θ=1/tan θ
Consider a right triangle with the three sides labeled with respect to an acute angle θ.
LHS * 1/sin θ=RHS* 1/sin θ
LHS * hyp/opp=RHS* hyp/opp
Rearrange equation
The tangent of an angle θ can be expressed as the ratio of the sine of θ to the cosine of θ.
tanθ = sinθ/cosθ
Similarly, the cotangent of θ can be expressed as the ratio of the cosine of θ to the sine of θ.
cotθ = cosθ/sinθ
Two proofs will be written for this identity, one using a right triangle and the other using a unit circle.
In a right triangle, the tangent of an angle θ is defined as the ratio of the length of the opposite side k to the length of the adjacent side l.
a/b=.a /m./.b /m.
k/m= sin(θ), l/m= cos(θ)
Consider a unit circle and an angle θ in standard position.
It is known that the point of intersection P of the terminal side of the angle and the unit circle has coordinates (cos θ , sin θ).
Draw a right triangle using the origin and P(cos θ , sin θ) as two of its vertices. The length of the hypotenuse is 1 and the lengths of the legs are sin θ and cos θ.
As shown previously, the tangent of a right triangle is defined as the ratio of the length of the opposite side — in this case, sin θ — to the length of the adjacent side, which here is cos θ. tan θ =sin θ/cos θ ✓
Two more proofs will be written for this identity, one of them using just a right triangle and the other using a unit circle.
In a right triangle, the cotangent of an angle θ is defined as the ratio of the length of the adjacent side l to the length of the opposite side k.
a/b=.a /m./.b /m.
l/m= cos(θ), k/m= sin(θ)
LHS * cos θ/sin θ=RHS* cos θ/sin θ
.LHS /tan θ.=.RHS /tan θ.
cot(θ) = 1/tan(θ)
Rearrange equation
For any angle θ, the following trigonometric identities hold true.
Definition | Substitute | Simplify | |
---|---|---|---|
sin θ | Length of oppositeside to∠ θ/Hypotenuse | opp/1 | opp |
cos θ | Length of adjacentside to∠ θ/Hypotenuse | adj/1 | adj |
It can be seen that if the hypotenuse of a right triangle is 1, the sine of an acute angle is equal to the length of its opposite side. Similarly, the cosine of the angle is equal to the length of its adjacent side.
By the Pythagorean Theorem, the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Therefore, for the above triangle, the sum of the squares of sin θ and cos θ is equal to the square of 1.
.LHS /cos^2θ.=.RHS /cos^2θ.
Write as a sum of fractions
a/a=1
Write as a power
a^m/b^m=(a/b)^m
sin θ/cos θ=tan θ
1/cos θ=sec θ
Commutative Property of Addition
.LHS /sin^2θ.=.RHS /sin^2θ.
Write as a sum of fractions
a/a=1
Write as a power
a^m/b^m=(a/b)^m
cos θ/sin θ=cot θ
1/sin θ=csc θ
1+cot ^2 θ=csc ^2 θ
The first identity can be shown using the unit circle and the Pythagorean Theorem. Consider a point (x,y) on the unit circle in the first quadrant, corresponding to the angle θ. A right triangle can be constructed with θ.
By the Pythagorean Theorem, the sum of the squares of x and y equals 1. x^2 + y^2 = 1 In fact, this is true not only for points in the first quadrant, but for every point on the unit circle. Recall that, for points (x,y) on the unit circle corresponding to angle θ, it is known that x = cos θ and that y = sin θ. By substituting these expressions into the equation, the first identity can be obtained.
The function y = sin(x) has odd symmetry, and y = cos(x) has even symmetry, which can be seen from their graphs. As a result, the corresponding identities hold true.
sin(- θ) = - sinθ cos(- θ) = cosθ tan(- θ) = - tanθ
The identities will be proven using a unit circle. Consider an arbitrary ∠ θ on a unit circle. Let A be the point that the angle forms on the circle.
Recall that the values on the x-axis are represented by cosine and the value on the y-axis are represented by sine. Therefore, the coordinates of A are (cosθ,sinθ).
Additionally, after the reflection, the angle between A', the origin, and the x-axis is - θ. This means that the lengths of the newly created horizontal and vertical segments are cos(- θ) and sin(- θ). Therefore, the coordinates of A' can also be written as (cos(- θ),sin(- θ)).
This way there are two pairs of coordinates of the same point A'. This allows to conclude the following. A'( cos(- θ), sin(- θ)) &and A'( cosθ, - sinθ) & ⇓ cos(- θ)&= cos θ sin(- θ)&= - sinθ
tan(θ)=sin(θ)/cos(θ)
sin(- θ)=- sin(θ)
cos(- θ)=cos(θ)
Put minus sign in front of fraction
sin(θ)/cos(θ)=tan(θ)
For any angle θ, the following trigonometric identities hold true.
sin ( π/2 - θ ) = cosθ
\cos\left(\dfrac \pi 2 - \theta\right) = \sin\theta
\tan\left(\dfrac \pi 2 - \theta\right) = \cot\theta
Consider a right triangle. The measure of its right angle is 90^(∘) or π2 radians. Let θ be the radian measure of one of the acute angles. Since the sum of two acute angles in a right triangle is π2, the measure of the third acute angle must be π2-θ.
Let also a, b, and c represent the side lengths of the triangle. In this case, cosine of θ can be expressed as the ratio of the lengths of the angle's adjacent side and the hypotenuse. cosθ = b/c At the same time, sine of the opposite angle can be expressed as the ratio of the lengths of the angle's opposite side and the hypotenuse. sin ( π/2 - θ ) = b/c Since the right-hand sides of the equations are equal, by the Transitive Property of Equality, the left-hand sides are also equal.
sin ( π/2 - θ ) = cosθ
This identity is true for all angles, not just those that make it possible to construct a right triangle. Using similar reasoning, the corresponding identities for cosine and tangent can be proven.
sin(α-β)=sin(α)cos(β)-cos(α)sin(β)
\ifnumequal{90}{0}{\sin\left(0\right)=0}{}\ifnumequal{90}{30}{\sin\left(\dfrac{\pi}{6}\right)=\dfrac{1}{2}}{}\ifnumequal{90}{45}{\sin\left(\dfrac{\pi}{4}\right)=\dfrac{\sqrt{2}}{2}}{}\ifnumequal{90}{60}{\sin\left(\dfrac{\pi}{3}\right)=\dfrac{\sqrt{3}}{2}}{}\ifnumequal{90}{90}{\sin\left(\dfrac{\pi}{2}\right)=1}{}\ifnumequal{90}{120}{\sin\left(\dfrac{2\pi}{3}\right)=\dfrac{\sqrt{3}}{2}}{}\ifnumequal{90}{135}{\sin\left(\dfrac{3\pi}{4}\right)=\dfrac{\sqrt{2}}{2}}{}\ifnumequal{90}{150}{\sin\left(\dfrac{5\pi}{6}\right)=\dfrac{1}{2}}{}\ifnumequal{90}{180}{\sin\left(\pi\right)=0}{}\ifnumequal{90}{210}{\sin\left(\dfrac{7\pi}6\right)=\text{-} \dfrac 1 2}{}\ifnumequal{90}{225}{\sin\left(\dfrac{5\pi}{4}\right)=\text{-} \dfrac {\sqrt{2}} {2}}{}\ifnumequal{90}{240}{\sin\left(\dfrac{4\pi}3\right)=\text{-} \dfrac {\sqrt 3}2}{}\ifnumequal{90}{270}{\sin\left(\dfrac{3\pi}{2}\right)=\text{-} 1}{}\ifnumequal{90}{300}{\sin\left(\dfrac{5\pi}3\right)=\text{-} \dfrac {\sqrt 3}2}{}\ifnumequal{90}{315}{\sin\left(\dfrac{7\pi}4\right)=\text{-} \dfrac {\sqrt{2}} {2}}{}\ifnumequal{90}{330}{\sin\left(\dfrac{11\pi}6\right)=\text{-} \dfrac 1 2}{}\ifnumequal{90}{360}{\sin\left(2\pi\right)=0}{}
a * 1=a
\ifnumequal{90}{0}{\cos\left(0\right)=1}{}\ifnumequal{90}{30}{\cos\left(\dfrac{\pi}{6}\right)=\dfrac{\sqrt{3}}{2}}{}\ifnumequal{90}{45}{\cos\left(\dfrac{\pi}{4}\right)=\dfrac{\sqrt{2}}{2}}{}\ifnumequal{90}{60}{\cos\left(\dfrac{\pi}{3}\right)=\dfrac{1}{2}}{}\ifnumequal{90}{90}{\cos\left(\dfrac{\pi}{2}\right)=0}{}\ifnumequal{90}{120}{\cos\left(\dfrac{2\pi}{3}\right)=\text{-} \dfrac{1}{2}}{}\ifnumequal{90}{135}{\cos\left(\dfrac{3\pi}{4}\right)=\text{-} \dfrac{\sqrt{2}}{2}}{}\ifnumequal{90}{150}{\cos\left(\dfrac{5\pi}{6}\right)=\text{-} \dfrac{\sqrt{3}}{2}}{}\ifnumequal{90}{180}{\cos\left(\pi\right)=\text{-} 1}{}\ifnumequal{90}{210}{\cos\left(\dfrac{7\pi}6\right)=\text{-} \dfrac{\sqrt 3}2}{}\ifnumequal{90}{225}{\cos\left(\dfrac{5\pi}{4}\right)=\text{-} \dfrac {\sqrt{2}} {2}}{}\ifnumequal{90}{240}{\cos\left(\dfrac{4\pi}3\right)=\text{-} \dfrac {1}2}{}\ifnumequal{90}{270}{\cos\left(\dfrac{3\pi}{2}\right)=0}{}\ifnumequal{90}{300}{\cos\left(\dfrac{5\pi}3\right)=\dfrac{1}2}{}\ifnumequal{90}{315}{\cos\left(\dfrac{7\pi}4\right)=\dfrac {\sqrt{2}} {2}}{}\ifnumequal{90}{330}{\cos\left(\dfrac{11\pi}6\right)=\dfrac{\sqrt 3}2}{}\ifnumequal{90}{360}{\cos\left(2\pi\right)=1}{}
Zero Property of Multiplication
Identity Property of Addition
sin ( π/2 - θ )=cos(θ)
To evaluate trigonometric functions of the sum of two angles, the following identities can be applied.
sin(x+y) &= sin x cos y + cos x sin y [0.8em] cos(x+y) &= cos x cos y - sin x sin y [0.8em] tan(x+y) &= tan x + tan y/1 - tan x tan y
There are also similar identities for the difference of two angles.
sin(x-y) &= sin x cos y - cos x sin y [0.8em] cos(x-y) &= cos x cos y + sin x sin y [0.8em] tan(x-y) &= tan x - tan y/1 + tan x tan y
Let △ AFD be a right triangle with hypotenuse 1 and an acute angle with measure x+y.
By definition, the sine of an angle is the ratio between the lengths of the opposite side and the hypotenuse. sin(x+y) = DF/1 ⇓ DF = sin(x+y) The idea now is to rewrite DF in terms of sin x, sin y, cos x, and cos y. To do it, draw a ray so that ∠ A is divided into two angles with measures x and y. Let C be a point on this ray such that △ ACD and △ ABC are right triangles.
AC= cos x
LHS * cos x=RHS* cos x
Rearrange equation
By the Third Angle Theorem, it is known that ∠ GAF ≅ ∠ GDC. Therefore, m∠ GDC = y.
Since the purpose is to rewrite DF, plot a point E on DF such that EC ∥ AB. This way a rectangle ECBF is formed. The opposite sides of a rectangle have the same length, so EF and CB are equal. Also, CE⊥ DF makes △ CED a right triangle.
Consequently, EF = cos x sin y and DE can be written in terms of sin x and cos y using the cosine ratio. cos y = DE/sin x ⇓ DE = sin x cos y Finally, by the Segment Addition Postulate, DF is equal to the sum of DE and EF. All these lengths have been rewritten in terms of the sine and cosine of x and y. DF = DE+ EF ⇓ sin(x+y) = sin x cos y + cos xsin y This concludes the proof of the first identity. The other identities can be proven using similar reasoning.
Consider the following process for calculating the exact value of sin 120^(∘).
Rewrite 120^(∘) as 90^(∘)+30^(∘)
sin(x+y) = sin x cos y + cos x sin y
Substitute values
1* a=a
Zero Property of Multiplication
Identity Property of Addition
The double-angle identities materialize when two angles with the same measure are substituted into the angle sum identities.
cl Sine: & sin 2θ=2sin θ cos θ [0.7em] & cos 2θ =cos^2 θ - sin^2 θ Cosine: & cos 2θ = 2cos^2 θ - 1 & cos 2θ = 1 - 2sin^2 θ [0.2cm] Tangent: & tan 2θ =2tanθ/1-tan^2 θ
Start by writing the Angle Sum Identity for sine and cosine. sin(x + y) = sin x cos y + cos x sin y cos(x + y) = cos x cos y - sin x sin y
Let x=θ and y=θ. With this, x+y becomes 2θ. Then, these two formulas can be rewritten in terms of θ. sin 2θ = sinθ cosθ + cosθ sinθ cos 2θ = cosθ cosθ - sinθ sinθ
Approaching the first equation, the Commutative Property of Multiplication can be applied to the second term of its right-hand side. Then, by adding the terms on the right-hand side of this equation, the formula for sin 2θ is obtained.
sin 2θ = sinθ cosθ + sinθ cosθ ⇓ sin 2θ = 2sinθ cosθ ✓
Approaching the second equation, the Product of Powers Property can be used to rewrite its right-hand side. By doing this, the first identity for the cosine of the double of an angle is obtained.
cos 2θ = cosθ cosθ - sinθ sinθ ⇓ cos 2θ = cos^2θ - sin^2θ ✓
sin^2 θ= 1-cos^2 θ
Distribute -1
Add terms
Write as a difference of fractions
Cross out common factors
Cancel out common factors
a^m/b^m=(a/b)^m
a* b/c=a*b/c
sin(θ)/cos(θ)=tan(θ)
To calculate the exact value of cos 120^(∘), these steps can be followed.
Rewrite 120^(∘) as 2* 60^(∘)
cos 2θ = 2cos^2 θ - 1
cos 60^(∘) = 1/2
The half-angle identities are special cases of angle difference identities. To evaluate trigonometric functions of half an angle, the following identities can be applied.
sin θ/2 &= ±sqrt(1 - cosθ/2) [0.75em] cos θ/2 &= ±sqrt(1 + cosθ/2) [0.75em] tan θ/2 &= ±sqrt(1 - cosθ/1 + cosθ)
The sign of each formula is determined by the quadrant where the angle θ2 lies.
These identities are useful when finding the exact value of the sine, cosine, or tangent at a given angle.
First, write two of the Double-Angle Identities for cosine. cos 2x &= 1 - 2sin^2 x cos 2x &= 2cos^2 x - 1
LHS-1=RHS-1
.LHS /(-2).=.RHS /(-2).
Rearrange equation
Put minus sign in front of fraction
-(b-a)=a-b
sqrt(LHS)=sqrt(RHS)
sin θ/2 &= ±sqrt(1-cos(2* θ2)/2) & ⇓ sin θ/2 &= ±sqrt(1-cosθ/2) ✓
LHS+1=RHS+1
.LHS /2.=.RHS /2.
Rearrange equation
sqrt(LHS)=sqrt(RHS)
cos θ/2 &= ±sqrt(1+cos (2* θ2)/2) & ⇓ cos θ/2 &= ±sqrt(1+cosθ/2) ✓
sin θ2= ±sqrt(1 - cosθ/2), cos θ/2= ±sqrt(1 + cosθ/2)
sqrt(a)/sqrt(b)=sqrt(a/b)
.a /b./.c /d.=a/b*d/c
Cross out common factors
Cancel out common factors
Multiply fractions
Consider the calculation of the exact value of cos 15^(∘).
Rewrite 15^(∘) as 30^(∘)/2
cos θ/2=sqrt(1+cos θ/2)
cos 30^(∘) =sqrt(3)/2