Trigonometric Identities & Equations: Test Review

Everything for the Unit 5 test on one page: the key facts from all 5 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Basic Trigonometric Identities

$csc θ = \frac{1}{sin θ}$ , $sec θ = \frac{1}{cos θ}$ , $cot θ = \frac{1}{tan θ} = \frac{cos θ}{sin θ}$ .
$tan θ = \frac{sin θ}{cos θ}$ — undefined wherever $cos θ = 0$ , i.e. at $θ = \frac{π}{2} + kπ$ .
$sin^{2} θ + cos^{2} θ = 1$ , and its rearrangements $sin^{2} θ = 1 - cos^{2} θ$ and $cos^{2} θ = 1 - sin^{2} θ$ .
Dividing by $cos^{2} θ$ gives $tan^{2} θ + 1 = sec^{2} θ$ ; dividing by $sin^{2} θ$ gives $1 + cot^{2} θ = csc^{2} θ$ .
$cos (- θ) = cos θ$ (even); $sin (- θ) = - sin θ$ and $tan (- θ) = - tan θ$ (odd).
$sin (\frac{π}{2} - θ) = cos θ$ and $cos (\frac{π}{2} - θ) = sin θ$ .

Compound Angle Formulas

$sin (A \pm B) = sin A cos B \pm cos A sin B$ — mixed products, sign preserved.
$cos (A \pm B) = cos A cos B \mp sin A sin B$ — like products, sign FLIPPED.
$tan (A \pm B) = \frac{tan A \pm tan B}{1 \mp tan A tan B} .$
$sin \frac{π}{12} = \frac{6 - 2}{4}$ and $cos \frac{π}{12} = \frac{6 + 2}{4}$ .
$sin \frac{5 π}{12} = \frac{6 + 2}{4}$ and $cos \frac{5 π}{12} = \frac{6 - 2}{4}$ .
Read every formula backwards too: $sin X cos Y + cos X sin Y$ collapses to $sin (X + Y)$ .

Double Angle Formulas

$sin 2 θ = 2 sin θ cos θ$ .
$cos 2 θ = cos^{2} θ - sin^{2} θ = 2 cos^{2} θ - 1 = 1 - 2 sin^{2} θ$ — all three are equal.
$tan 2 θ = \frac{2 tan θ}{1 - tan ^{2} θ}$ .
The formulas generalize: $sin 6 x = 2 sin 3 x cos 3 x$ , because the half-angle is whatever is half of the angle in front of you.
Rearranged for proofs: $1 + cos 2 θ = 2 cos^{2} θ$ and $1 - cos 2 θ = 2 sin^{2} θ$ .

Proving Identities & Solving Equations

In a proof, work ONE side only. Moving terms across the equals sign assumes the result and is circular.
Default proof strategy: convert to sine and cosine, then look for a Pythagorean substitution, then combine over a common denominator, then factor.
In an equation you MAY rearrange freely. Get everything to one side, reduce to a single trig ratio, factor, then solve each factor.
Reference angle first, quadrants second: find the acute angle from the positive value, then place it wherever the CAST rule matches the sign.
On $0 \leq x < 2 π$ , a sine or cosine equation normally has TWO solutions, and $tan x = k$ has two solutions exactly $π$ apart.
Never divide both sides by a trig ratio — factor instead, or you will delete solutions.

Trig Equations with Multiple Angles and Applications

For $sin (k x) = c$ on $0 \leq x < 2 π$ , solve $sin θ = c$ on $0 \leq θ < 2 kπ$ , then divide every angle by $k$ .
For $θ = k x + b$ , transform the interval the same way: multiply by $k$ , then add $b$ ; undo in reverse order at the end.
A whole-number $k$ normally gives $2 k$ solutions for sine or cosine with $∣ c ∣ < 1$ ; a fraction such as $\frac{x}{2}$ shrinks the interval and can leave one solution or none.
If an equation mixes $2 x$ and $x$ , expand the double angle first: $sin 2 x = 2 sin x cos x$ ; pick $cos 2 x = 2 cos^{2} x - 1$ or $1 - 2 sin^{2} x$ to match the other ratio.
Recognize compound forms: $sin x cos a + cos x sin a = sin (x + a)$ and $cos x cos a - sin x sin a = cos (x + a) .$
In applied problems the interval comes from the context (one day, one revolution); state it first and interpret every answer with units.

Where marks get lost

Reading $sin^{2} θ$ as $sin (θ^{2})$ . The exponent belongs to the whole ratio: $sin^{2} θ = (sin θ)^{2}$ . (Basic Trigonometric Identities)
Pairing the reciprocals wrongly — writing $csc θ = \frac{1}{cos θ}$ . Cosecant goes with SINE and secant goes with COSINE; the mnemonic is that the third letter of each reciprocal name tells you its partner. (Basic Trigonometric Identities)
Confusing $csc θ$ (reciprocal of sine) with $sin^{- 1} θ$ (the inverse function). They are completely different operations that happen to share the $^{- 1}$ notation. (Basic Trigonometric Identities)
Taking the square root in $sin θ = \pm 1 - cos^{2} θ$ and keeping only the positive root. The quadrant decides the sign, and half the time it is negative. (Basic Trigonometric Identities)
Assuming an identity holds everywhere with no exceptions. $tan θ = \frac{sin θ}{cos θ}$ is undefined at $θ = \frac{π}{2}$ , so the identity is only claimed on the domain where both sides exist. (Basic Trigonometric Identities)
Distributing: writing $cos (A + B) = cos A + cos B$ . There is no distributive law for trig functions — $cos$ is a function name, not a factor. (Compound Angle Formulas)
Getting the cosine sign backwards. $cos (A + B)$ has a MINUS between the products and $cos (A - B)$ has a PLUS. Sine is the one that keeps its sign. (Compound Angle Formulas)
Mixing the ratios in the cosine formula — writing $cos A sin B - sin A cos B$ . The cosine formula pairs like with like: cosine with cosine, sine with sine. (Compound Angle Formulas)
Decomposing an angle incorrectly, e.g. claiming $\frac{π}{12} = \frac{π}{6} - \frac{π}{4}$ . Convert to twelfths and check: that difference is $- \frac{π}{12}$ , not $\frac{π}{12}$ . (Compound Angle Formulas)
When given $sin A$ and asked for $cos (A + B)$ , forgetting that you must first find $cos A$ and $sin B$ — and that the quadrant, not the algebra, decides their signs. (Compound Angle Formulas)
Writing $sin 2 θ = 2 sin θ$ . Doubling the angle is not doubling the ratio — the correct formula is $2 sin θ cos θ$ , and the extra cosine changes everything. (Double Angle Formulas)
Dropping the $2$ in front: writing $sin 2 θ = sin θ cos θ$ . The two identical terms from the compound formula add, they do not merge. (Double Angle Formulas)
Mixing up the two one-sided forms of $cos 2 θ$ and writing $2 sin^{2} θ - 1$ or $1 - 2 cos^{2} θ$ . Both are wrong: the COSINE form ends in $- 1$ , the SINE form starts with $1 -$ . (Double Angle Formulas)
Using $tan 2 θ = \frac{2 tan θ}{1 - tan θ}$ without the square. The denominator must be $1 - tan^{2} θ$ . (Double Angle Formulas)
Assuming $2 θ$ lies in the same quadrant as $θ$ and forcing the signs of $sin 2 θ$ and $cos 2 θ$ to match. Use the quadrant only to fix $sin θ$ and $cos θ$ , then let the formulas produce the doubled signs. (Double Angle Formulas)
Cross-multiplying or moving terms across the equals sign inside a PROOF. That assumes the identity is true, which is the thing being proven, and earns no marks even if the last line is correct. (Proving Identities & Solving Equations)
Stopping after the calculator's first answer. $cos x = - \frac{1}{2}$ on $0 \leq x < 2 π$ has solutions $\frac{2 π}{3}$ AND $\frac{4 π}{3}$ ; reporting only one loses half the marks. (Proving Identities & Solving Equations)
Dividing both sides of an equation by a trig ratio. Dividing $sin 2 x = sin x$ by $sin x$ deletes the solutions $x = 0$ and $x = π$ . Always factor. (Proving Identities & Solving Equations)
Forgetting to widen the interval for a multiple angle. For $sin 2 x = k$ on $0 \leq x < 2 π$ you must solve over $0 \leq 2 x < 4 π$ , which usually gives four solutions for $x$ . (Proving Identities & Solving Equations)
Taking the reference angle from a negative value. Always feed the POSITIVE version of the number into the inverse function, then use CAST to decide the quadrants and signs. (Proving Identities & Solving Equations)
Solving $sin 2 x = c$ over $[0, 2 π)$ and then halving, which loses half of the solutions. (Trig Equations with Multiple Angles and Applications)
Forgetting to divide by $k$ at the end and reporting values of $2 x$ as if they were $x$ . (Trig Equations with Multiple Angles and Applications)
Keeping the angle $\frac{11 π}{6}$ for $cos \frac{x}{2} = \frac{3}{2}$ , even though $\frac{x}{2}$ only runs over $[0, π)$ ; it gives $x = \frac{11 π}{3}$ , outside the interval. (Trig Equations with Multiple Angles and Applications)
Shifting the interval in the wrong order for $2 x - \frac{π}{3}$ : multiply by 2 first, then subtract $\frac{π}{3}$ . (Trig Equations with Multiple Angles and Applications)
Dividing $sin 2 x + cos x = 0$ by $cos x$ instead of factoring $cos x (2 sin x + 1) = 0$ , which deletes $x = \frac{π}{2}$ and $\frac{3 π}{2}$ . (Trig Equations with Multiple Angles and Applications)

Practice test

25 questions from across the unit. Each one shows the lesson it comes from.

Basic Trigonometric Identities
1
Which of these statements is an IDENTITY rather than an equation?
Basic Trigonometric Identities
2
If $sin θ = 0.6$ and $θ$ lies in quadrant I, find $cos θ$ .
Basic Trigonometric Identities
3
Simplify $sin θ csc θ$ to a single number.
Basic Trigonometric Identities
4
Which statement about even and odd identities is correct?
Basic Trigonometric Identities
5
Match each identity to the family it belongs to.
$sec θ = \frac{1}{cos θ}$
$cot θ = \frac{cos θ}{sin θ}$
$1 + cot^{2} θ = csc^{2} θ$
$tan (- θ) = - tan θ$
$cos (\frac{π}{2} - θ) = sin θ$
Compound Angle Formulas
6
Which expansion of $cos (A + B)$ is correct?
Compound Angle Formulas
7
Which decomposition correctly produces $\frac{π}{12}$ ?
Compound Angle Formulas
8
$tan \frac{5 π}{12} = 2 + 3$ . Give its value to three decimal places.
Compound Angle Formulas
9
True or false: $cos (A + B) = cos A + cos B$ for all $A$ and $B$ .
Compound Angle Formulas
10
Match each expression to the single trigonometric ratio it collapses to.
1. A. $cos 3 x$
2. B. $cos 5 x$
3. C. $sin 3 x$
4. D. $sin 5 x$
$sin 4 x cos x + cos 4 x sin x$
$cos 4 x cos x - sin 4 x sin x$
$cos 4 x cos x + sin 4 x sin x$
$sin 4 x cos x - cos 4 x sin x$
Double Angle Formulas
11
Which is the correct double angle formula for sine?
Double Angle Formulas
12
Given $sin θ = \frac{3}{5}$ with $θ$ in quadrant II, find $cos 2 θ$ as a decimal.
Double Angle Formulas
13
Given $sin θ = \frac{3}{5}$ with $θ$ in quadrant I, find $sin 2 θ$ as a decimal.
Double Angle Formulas
14
Simplify $1 - 2 sin^{2} 4 x$ to a single trigonometric ratio.
Double Angle Formulas
15
Match each expression to its simplified form.
1. A. $cos 2 θ$
2. B. $sin 2 θ$
3. C. $tan 2 θ$
4. D. $2 cos^{2} θ$
5. E. $2 sin^{2} θ$
$2 sin θ cos θ$
$1 + cos 2 θ$
$1 - cos 2 θ$
$cos^{2} θ - sin^{2} θ$
$\frac{2 tan θ}{1 - tan ^{2} θ}$
Proving Identities & Solving Equations
16
Why is cross-multiplying not allowed when proving an identity?
Proving Identities & Solving Equations
17
Which list gives all solutions of $sin x = \frac{3}{2}$ on $0 \leq x < 2 π$ ?
Proving Identities & Solving Equations
18
How many solutions does $tan x = 1$ have on $0 \leq x < 2 π$ ?
Proving Identities & Solving Equations
19
When solving $sin 2 x = sin x$ , why must you factor rather than divide both sides by $sin x$ ?
Proving Identities & Solving Equations
20
Match each equation to the pair of quadrants containing its solutions on $0 \leq x < 2 π$ .
$sin x = 0.4$
$sin x = - 0.4$
$cos x = 0.4$
$cos x = - 0.4$
Trig Equations with Multiple Angles and Applications
21
How many solutions does $sin 3 x = \frac{1}{2}$ have on $0 \leq x < 2 π$ ?
Trig Equations with Multiple Angles and Applications
22
Solve $cos \frac{x}{2} = \frac{3}{2}$ on $0 \leq x < 2 π$ . Type the single solution, e.g. pi/4.
Trig Equations with Multiple Angles and Applications
23
Find the largest solution of $2 cos 2 x = - 1$ on $0 \leq x < 2 π$ , in radians to 3 decimal places.
Trig Equations with Multiple Angles and Applications
24
What is the best first step for $cos 2 x + cos x = 0$ ?
Trig Equations with Multiple Angles and Applications
25
$sin x cos \frac{π}{3} + cos x sin \frac{π}{3} = \frac{1}{2}$ simplifies to which equation?

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…