A triangle has sides of length 5 and 7 with included angle 60^°. What is the length of the third side?

Worked solution for TMUA-P1-MM4-005 is revealed on this page behind the solution details.

Why isn't the answer to TMUA-P1-MM4-005 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P1-MM4-005 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P1-MM4-005?

No — calculators are banned in the TMUA. TMUA-P1-MM4-005 is set for a no-calculator target of about 210 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P1-MM4-005?

TMUA-P1-MM4-005 is Paper P1 · Trigonometry (specification code MM4), level 2. Drill more of the same topic from this page.

TMUA-P1-MM4-005 — Paper P1 · Trigonometry · Level 2

P1MM4Level 2~210s target

Spec: MM4 · No calculator · ~210s

A triangle has sides of length

5

and

7

with included angle

60^\circ

. What is the length of the third side?

Options

$\sqrt{109}$
$\sqrt{74}$
$\sqrt{39}$
$\sqrt{14}$
$13$

Letters only — the answer key is hidden. Work it out, then check the solution below.

Hint (check trope)

No written hint yet — check the trope: cosine-rule, sign-trap. Name the trap first, work the stem by hand, then open the solution below.

Show solution

Cosine rule:

c^2 = 25 + 49 - 2(5)(7)\cos 60^\circ = 74 - 35 = 39

, so

c = \sqrt{39}

. Trap:

\sqrt{109}

adds the

35

instead of subtracting — plausible if you misremember the sign in the cosine rule.

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Browse more

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Keep practising

Same topic · same level

P1MM4Level 2

TMUA-P1-MM4-006

Find all values of θ with 0 θ < 360^° satisfying 2θ - 1 = 0.

Trigonometry

Open TMUA-P1-MM4-006 →

Same trap: sign-trap

P1MM7Level 2

TMUA-P1-MM7-003

The curve y = 2x - 6 lies below the x -axis between x = 1 and x = 3. What is the area enclosed by the curve, the x -axis…

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Open TMUA-P1-MM7-003 →

Step up a level

P1MM4Level 3

TMUA-P1-MM4-011

A triangle has sides of length 4, 5 and 6. What is the cosine of its largest angle?

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Open TMUA-P1-MM4-011 →

TMUA-P1-MM4-005 — Paper P1 · Trigonometry · Level 2

Options

Keep practising

Master Trigonometry

TMUA-P1-MM4-005 — Paper P1 · Trigonometry · Level 2

Options

Keep practising

Master Trigonometry