B. Trigonometric Functions · Trigonometric Identities and EquationsEquivalent Expressions from Transformations and Symmetry
B3.1About 70 minutes
Learning goals- I can use horizontal translations of graphs to recognize equivalent expressions such as cos(x + π/2) = −sin x.
- I can use symmetry on the unit circle to write sin(π − x), cos(π + x), sin(2π − x) and similar expressions in terms of sin x or cos x.
- I can use periodicity and the even/odd properties of sine and cosine to simplify expressions.
- I can show that two expressions are not equivalent by finding one value where they differ, and check equivalence with graphing technology.
Words to know
periodic function
A function whose values repeat after a fixed horizontal distance, the period. sin x and cos x have period 2π; tan x has period π.
even function
A function with f(−x) = f(x) for every x; its graph is symmetric about the y-axis. cos x is even.
odd function
A function with f(−x) = −f(x) for every x; its graph has rotational symmetry about the origin. sin x and tan x are odd.
unit circle
The circle of radius 1 centred at the origin. The point at angle x in standard position is (cos x, sin x).
horizontal translation
A shift left or right. The graph of y = f(x + c) is the graph of y = f(x) shifted c units left (for c > 0).
One curve, many names
In the last lesson you saw that sin x and cos(π/2 − x) are two names for the same function. There are many more. Every trigonometric graph can be shifted, reflected or rotated onto itself or onto another trigonometric graph, and each of those moves gives an equivalent expression.
We will use three tools you met in the unit Radian Measure and Trigonometric Functions:
- Transformations of graphs (shifts and reflections).
- Symmetry of points on the unit circle.
- Periodicity and the even/odd properties of sine and cosine.
Tool 1: Periodicity and even/odd
Adding a full turn of 2π brings you back to the same point on the unit circle, so:
- sin(x + 2π) = sin x and cos(x + 2π) = cos x
- tan(x + π) = tan x (tangent repeats every π)
Going the other way around the circle (angle −x) reflects the point in the x-axis: the x-coordinate stays and the y-coordinate changes sign. So:
- cos(−x) = cos x (cosine is even)
- sin(−x) = −sin x (sine is odd)
- tan(−x) = −tan x (tangent is odd)
Tool 2: Symmetry on the unit circle
Let P(cos x, sin x) be the point at angle x. Reflecting or rotating P gives three partner points:
- π − x: reflect P in the y-axis. The x-coordinate changes sign: (−cos x, sin x).
- π + x: rotate P by half a turn. Both coordinates change sign: (−cos x, −sin x).
- 2π − x (same as −x): reflect P in the x-axis: (cos x, −sin x).
Reading the coordinates gives the identities in the table below. This is the related-acute-angle idea from Grade 11 (CAST), now written as identities in radians.
| Angle | sine | cosine | tangent |
|---|
| π − x | sin x | −cos x | −tan x |
| π + x | −sin x | −cos x | tan x |
| 2π − x | −sin x | cos x | −tan x |
| −x | −sin x | cos x | −tan x |
| π/2 − x | cos x | sin x | cot x |
| π/2 + x | cos x | −sin x | −cot x |
Tool 3: Transformations of graphs
The graph of y = cos(x + π/2) is the graph of y = cos x shifted π/2 to the left. The cosine minimum at x = π moves to x = π/2. The result starts at 0, goes down to −1 at π/2, and comes back up to 1 at 3π/2. That is exactly the graph of y = sin x reflected in the x-axis, which is y = −sin x. So cos(x + π/2) = −sin x.
In the same way, sliding y = sin x left by π/2 puts its peak at x = 0, where the cosine peak is: sin(x + π/2) = cos x.
Worked exampleExample 1: Using a shift
Use a transformation to show that sin(x − π/2) = −cos x. Then check one value.
- y = sin(x − π/2) is y = sin x shifted π/2 to the right.
- The sine peak at x = π/2 moves to x = π, and the sine minimum at 3π/2 moves to 2π. At x = 0 the shifted graph has the value sin(−π/2) = −1.
- A curve with value −1 at 0, a peak of 1 at π and −1 at 2π is the cosine curve flipped upside down: y = −cos x.
- Check x = π/3: sin(π/3 − π/2) = sin(−π/6) = −1/2, and −cos(π/3) = −1/2 ✓.
Worked exampleExample 2: Exact values with symmetry
Find the exact value of sin(7π/6) and cos(5π/3) by writing each angle as π + x or 2π − x.
- 7π/6 = π + π/6, so sin(7π/6) = −sin(π/6) = −1/2.
- 5π/3 = 2π − π/3, so cos(5π/3) = cos(π/3) = 1/2.
sin(7π/6) = −1/2 and cos(5π/3) = 1/2
Worked exampleExample 3: Simplify a combination
Simplify sin(π − x) + sin(π + x) + cos(2π − x).
- sin(π − x) = sin x (reflection in the y-axis keeps the y-coordinate).
- sin(π + x) = −sin x (half turn changes both signs).
- cos(2π − x) = cos x (reflection in the x-axis keeps the x-coordinate).
- Add: sin x − sin x + cos x.
Worked exampleExample 4: Proving NOT equivalent
Your friend claims that cos(x + π/2) = sin x. Show that this is false.
- To show two expressions are not equivalent, one value of x where they differ is enough. This is called a counterexample.
- Try x = π/2: cos(π/2 + π/2) = cos π = −1, but sin(π/2) = 1.
- −1 ≠ 1, so the claim is false. (The correct statement is cos(x + π/2) = −sin x.)
The expressions are not equivalent; x = π/2 is a counterexample.
Watch out for x = 0
When you test a claim, avoid x = 0 as your only test. Many false claims happen to work at 0. For example, cos(x + π/2) and sin x are both 0 at x = 0, even though they are not equivalent. Test with values such as π/6, π/4, π/3 or 1, and use graphing technology: if two graphs separate anywhere, the expressions are not equivalent.
Just for funSine is odd and cosine is even. That is not gossip about their personalities: it just means sin(−x) = −sin x and cos(−x) = cos x. Tangent is odd too, which makes cosine the only even one of the three main ratios.
Hands-on activityA paper-plate unit circle
You need: a paper plate (or a circle traced around a large lid on paper), a ruler, a pencil and a protractor.
- Draw the x- and y-axes through the centre. Call the radius 1 unit.
- Mark a point P on the circle about 35° (about 0.61 rad) up from the positive x-axis. Measure its coordinates with the ruler as a fraction of the radius. (If the radius is 10 cm, a point 8.2 cm across and 5.7 cm up is (0.82, 0.57).)
- Fold the plate along the y-axis and press P through to mark its mirror image. This is the point for π − x. Measure its coordinates.
- Unfold, fold along the x-axis, and mark the image of P. This is the point for 2π − x.
- Rotate the plate half a turn about its centre (or draw a line from P through the centre to the far side). This is π + x.
What to look for: with x ≈ 0.61 rad, cos x ≈ 0.82 and sin x ≈ 0.57. Your four points should be close to (0.82, 0.57), (−0.82, 0.57), (−0.82, −0.57) and (0.82, −0.57). The sizes stay the same; only the signs change.
Your turn
1Use the unit circle to simplify cos(π − x).
Reflecting in the y-axis changes the sign of the x-coordinate: cos(π − x) = −cos x.
2Simplify sin(x + 2π) − sin(−x).
sin(x + 2π) = sin x and sin(−x) = −sin x. So sin x − (−sin x) = 2 sin x.
3Write sin(π/2 + x) in terms of a single ratio of x.
Shifting sine left by π/2 gives cosine: sin(π/2 + x) = cos x.
4Find the exact value of cos(4π/3) by writing 4π/3 = π + π/3.
cos(π + x) = −cos x, so cos(4π/3) = −cos(π/3) = −1/2.
5Find the exact value of tan(5π/4).
5π/4 = π + π/4 and tan(π + x) = tan x, so tan(5π/4) = tan(π/4) = 1.
6Show with a counterexample that sin(π + x) = sin x is false.
Try x = π/2: sin(3π/2) = −1, but sin(π/2) = 1. Not equal, so it is not an identity. (In fact sin(π + x) = −sin x.)
7Simplify cos(−x) + cos(π − x).
cos(−x) = cos x and cos(π − x) = −cos x. The sum is 0.
More practice: on your own
1Find the exact value of sin(11π/6) using 11π/6 = 2π − π/6.
sin(2π − x) = −sin x, so sin(11π/6) = −sin(π/6) = −1/2.
2Simplify sin(π − x) cos(π + x) + cos(π/2 − x) cos x.
sin(π − x) = sin x, cos(π + x) = −cos x, cos(π/2 − x) = sin x. So −sin x cos x + sin x cos x = 0.
3Describe a transformation of y = cos x that gives y = cos(x − π). Then write cos(x − π) in terms of cos x.
Shift y = cos x right by π. The peak at 0 moves to π, so the graph is upside down: cos(x − π) = −cos x. (Check x = 0: cos(−π) = −1 = −cos 0 ✓.)
4Is tan(π − x) = tan x an identity? Justify.
No. Try x = π/4: tan(3π/4) = −1 but tan(π/4) = 1. The correct identity is tan(π − x) = −tan x.
5Express cos(3π/2 − x) in terms of sin x or cos x. (Hint: 3π/2 − x = π + (π/2 − x).)
cos(π + y) = −cos y with y = π/2 − x, so cos(3π/2 − x) = −cos(π/2 − x) = −sin x. Check x = π/2: cos π = −1 and −sin(π/2) = −1 ✓.
Challenge
1Find all values of c with 0 ≤ c < 2π such that sin(x + c) = cos x is an identity. Explain.
We need the sine curve shifted left by c to land on the cosine curve. Shifting left by π/2 works, and because sine repeats every 2π, no other shift in [0, 2π) works (shifting by any other amount moves the peak away from x = 0). c = π/2 only.
2Simplify [sin(π − x) + sin(−x)] / cos(2π + x) and state any restrictions.
Numerator: sin x − sin x = 0. Denominator: cos x. The expression is 0, for every x where cos x ≠ 0, so x ≠ π/2 + kπ (k an integer).
3A student says: "cos(x + π) = cos(x − π) must be false, because one graph is shifted left and the other right." Is the student right? Explain with the period.
No. Shifting left by π and shifting right by π differ by a total shift of 2π, which is one full period of cosine. So the two graphs are identical: both equal −cos x. Check x = 0: cos π = −1 and cos(−π) = −1 ✓.
Quiz
1. Which expression is equivalent to cos(x + π/2)?
2. Which expression is equivalent to sin(π + x)?
4. What is the exact value of cos(7π/4)?
5. Which value of x is a counterexample showing that cos(π − x) = cos x is not an identity?
6. Two graphs drawn in radian mode lie exactly on top of each other over many periods. What can you conclude?
Big idea (B3.1): the same trigonometric curve can be written in many ways. Shifting, reflecting and using the symmetry of the unit circle turn one expression into an equivalent one, such as cos(x + π/2) = −sin x.
How to help: Ask your teen to sketch a unit circle and mark a point at a small angle x, then its mirror images. Ask: "Which coordinates change sign for π − x? For π + x?" If you have a graphing app (Desmos is free), graph both sides of a claimed identity together and see whether the curves match.
Common difficulties:
- Shifting the wrong way: y = f(x + c) moves the graph left for c > 0.
- Mixing up which reflection changes which sign: reflecting in the y-axis changes the x-coordinate (cosine); reflecting in the x-axis changes the y-coordinate (sine).
- Testing a claim only at x = 0, where many false claims happen to work.
- Forgetting that tangent repeats every π, not 2π.