What is a trigonometric identity ?

A trigonometric identity is a statement that is true for all values of θ or x in trigonometry.

True or False? Trigonometric identities can be used to prove double angle formulae.

True. Trigonometric identities can be used to prove further identities such as the double angle formulae.

How can you determine if a trigonometric ratio is positive or negative ?

You can determine if a trigonometric ratio is positive or negative by looking at its angle and identifying the quadrant of the unit circle in which it lies. Remember that the CAST diagram is labelled anti-clockwise from the 4th quadrant and identifies which ratios are positive, ( C os, A ll, S in and T an).

True or False? The symmetry properties of trigonometric graphs can be used to find all secondary values within a given interval .

True. The symmetry properties of trigonometric graphs can be used to find all secondary values within a given interval .

IBMathsDPAnalysis & Approaches (AA)SLFlashcards3. Geometry & TrigonometryTrigonometric Equations & Identities

Trigonometric Equations & Identities (DP IB Analysis & Approaches (AA))

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What is a trigonometric identity?

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What is a trigonometric identity?
A trigonometric identity is a statement that is true for all values of θ or x in trigonometry.
State the identity for $\tan θ$ .
The tan identity is $\tan θ = \frac{\sin θ}{\cos θ}$ .
This is given in your exam formula booklet.
State the Pythagorean identity.
The Pythagorean identity is $\sin^{2} θ + \cos^{2} θ = 1$ .
This is given in your exam formula booklet.
True or False?
Trigonometric identities can be used to prove double angle formulae.
True.
Trigonometric identities can be used to prove further identities such as the double angle formulae.
What is the double angle identity for the sine function?
The double angle identity for the sine function is: $\sin 2 θ = 2 \sin θ \cos θ$ .
This is given in your exam formula booklet.
What is the double angle identity for the cosine function?
The double angle identity for the cosine function is: $\cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} - 1 = 1 - 2 \sin^{2} θ$ .
This is given in your exam formula booklet.
How can you use the double angle identities to solve trigonometric equations such as $\sin 2 θ = \sin θ$ for $0 ° \leq θ \leq 360 °$ ?
You can the double angle identities to solve trigonometric equations by substituting in the expression that a double angle expression is equivalent to.
E.g. to solve $\sin 2 θ = \sin θ$ for $0 ° \leq θ \leq 360 °$ , substitute $2 \sin θ \cos θ$ for $\sin 2 θ$ , then simplify to end up with simple equations in terms of a single trig function to solve.
Given an equation that involves $\sin θ \cos θ$ , what expression should be substituted to simplify and solve the equation?
The double angle formula for the sine function is $\sin 2 θ = 2 \sin θ \cos θ$ .
Therefore, if an equation involves $\sin θ \cos θ$ , you should substitute $\frac{1}{2} \sin 2 θ$ for it in order to be able to simplify and solve the equation.
True or False?
To solve equations that contain $\sin 2 θ$ and either $\sin θ$ or $\cos θ$ , you will need to use the substitution $1 - \sin^{2} θ$ .
False.
To solve equations that contain $\sin 2 θ$ and either $\sin θ$ or $\cos θ$ , you will need to use the substitution $2 \sin θ \cos θ$ .
The substitution $1 - \sin^{2} θ$ should be used if the equation you are trying to solve includes $\cos 2 θ$ .
If an equation includes both $\cos 2 θ$ and either $\sin θ$ or $\cos θ$ , how do you decide which expression to substitute?
If an equation includes $\cos 2 θ$ look at what other terms are involved to decide which expression to substitute.
- If the equation involves $\cos θ$ , substitute $2 \cos^{2} θ - 1$ .
- If the equation involves $\sin θ$ , substitute $1 - 2 \sin^{2} θ$ .
How can you determine if a trigonometric ratio is positive or negative?
You can determine if a trigonometric ratio is positive or negative by looking at its angle and identifying the quadrant of the unit circle in which it lies.
Remember that the CAST diagram is labelled anti-clockwise from the 4th quadrant and identifies which ratios are positive, (Cos, All, Sin and Tan).
If you know two trig ratios, how can you use them to work out the third trig ratio?
If you know two trig ratios, you can use the tan identity, $\tan θ = \frac{\sin θ}{\cos θ}$ , to work out the third trig ratio.
Given the value of $\sin θ$ , how can you work out the value of $\cos θ$ ?
Given the value of $\sin θ$ , you can work out the value of $\cos θ$ by substituting the known value into the Pythagorean identity, $\sin^{2} θ + \cos^{2} θ = 1$
If you know that $\sin θ = \frac{a}{b}$ , where $a, b \in ℤ^{+}$ , how can you work out the values of $\cos θ$ and $\tan θ$ ?
If you know that $\sin θ = \frac{a}{b}$ , where $a, b \in ℤ^{+}$ , you can work out the values of $\cos θ$ and $\tan θ$ by:
- sketching a right-angled triangle with $a$ opposite $θ$ and $b$ on the hypotenuse,
- using Pythagoras’ theorem to find the value of the adjacent side,
- then using SOHCAHTOA to find the values of $\cos θ$ and $\tan θ$ .
What is a linear trigonometric equation?
A linear trigonometric equation is an equation that involves $\sin θ$ , $\cos θ$ or $\tan θ$ (but no powers of $\sin θ$ , $\cos θ$ or $\tan θ$ ).
It may be of the form $\sin x = k$ or $\sin (a x + b) = k$ .
Given an equation $\sin θ = k$ , how can you find a secondary value for possible values of $θ$ that solve the equation?
A secondary value for the solution to an equation $\sin θ = k$ can be found by subtracting $θ$ from 180º.
Further values can then be found by adding/subtracting 360º or $2 π$ radians to the primary value and the first secondary value.
Given an equation $\cos θ = k$ , how can you find a secondary value for possible values of $θ$ that solve the equation?
A secondary value for the solution to an equation $\cos θ = k$ can be found by subtracting $θ$ from 360º.
Further values can then be found by adding/subtracting 360º or $2 π$ radians to the primary value and the first secondary value.
What value can be added to or subtracted from a solution to an equation $\tan θ = k$ to find a secondary value for possible values of $θ$ that solve the equation?
180º or $π$ radians can be added to or subtracted from a solution to an equation $\tan θ = k$ to find a secondary solution.
True or False?
The symmetry properties of trigonometric graphs can be used to find all secondary values within a given interval.
True.
The symmetry properties of trigonometric graphs can be used to find all secondary values within a given interval.
How do you solve an equation of the form $\sin (a x + b) = k$ ?
To solve an equation of the form $\sin (a x + b) = k$ :
- Let $u = a x + b$ .
- Solve the function to find the primary value for $u$ .
- Transform the interval in the same way as the angle was transformed $(a x + b)$ .
- Find all secondary values in the transformed interval for $u$ .
- Undo the transformation on all values to convert back to solutions for $x$ .
What is a quadratic trigonometric equation?
A quadratic trigonometric equation is an equation that involves either $\sin^{2} θ$ , $\cos^{2} θ$ or $\tan^{2} θ$ .
Which trig identity is often used with quadratic trigonometric equations?
The trig identity, $\sin^{2} θ + \cos^{2} θ = 1$ , is often used to convert a trigonometric equation involving both $\sin θ$ and $\cos θ$ into a quadratic trigonometric equation in either $\sin θ$ or $\cos θ$ that can then be solved.
True or False?
A quadratic trigonometric equation will always generate at least one linear trigonometric equation that can be solved to give real solutions for the original equation.
False.
A quadratic trigonometric equation will not always generate at least one linear trigonometric equation that can be solved to give real solutions for the original equation.
E.g. an equation like $\sin^{2} x + 6 \sin x + 11 = 0$ has no real solutions for $\sin x$ (because $y^{2} + 6 y + 11$ has a negative discriminant).
Or an equation like $\sin^{2} x + \sin x - 6 = (\sin x - 2) (\sin x + 3) = 0$ gives the two linear equations $\sin x = 2$ and $\sin x = - 3$ . But those don't have real solutions for $x$ because, for any value of $x$ , the value of $\sin x$ can only be between -1 and 1.