Find the value of θ if (4θ) = 18°, where 4θ is an acute angle.

18°. Since sine equals cosine for complementary angles, their arguments must sum to 90°. We set up the equation 4θ + 18° = 90°. Subtracting 18° from both sides gives 4θ = 72°. Dividing by 4 gives θ = (72°)/(4) = 18°.

Grade 10 · Core Mathematics · Lesson 5 of 11

Solving Trigonometric Equations with Complementary Angles

A Grade 10 Core Mathematics CBE lesson from the sub-strand Trigonometry 1. In it, learners set up and solve algebraic equations for unknown angles using the complementary relationship sin A = cos(90° - A).

Strand Measurements and GeometrySub-strand Trigonometry 1Lesson code 10CM-24-L540 minutes · 2 slides

KICD learning outcome (b)

Relate sines and cosines of complementary angles.

By the end of this lesson, learners should be able to

Set up and solve algebraic equations for unknown angles using the complementary relationship sin A = cos(90° - A)
Apply complementary angle identities to determine sine and cosine values in right-angled triangles using side lengths and the Pythagorean theorem

Key vocabulary

Complementary Angles: Two acute angles whose measures add up to ninety degrees.; Angles measuring $35^{\circ}$ and $55^{\circ}$ are complementary angles because their sum is $90^{\circ}$ .
Angle Argument: The input angle value or algebraic expression placed inside a trigonometric function.; In the expression $\sin (5 θ)$ , the term $5 θ$ is the angle argument.
Pythagorean Theorem: The geometric rule stating that hypotenuse squared equals sum of squares of legs.; We used the Pythagorean theorem to calculate the missing adjacent side length of $24 cm$ .

Lesson preview

Slide 1 · Solving Trigonometric Equations via Complementary ArgumentsModel · 18 min

Figure 1: A flowchart showing the algebraic steps to solve a trigonometric equation by equating complementary angles to 90 degrees.

When two acute angles are complementary, their sum is $90^{\circ}$ , meaning the sine of one angle equals the cosine of the other. We can set up algebraic equations by equating the sum of the angle arguments to $90^{\circ}$ . For example, if $\sin 4 θ = \cos 50^{\circ}$ , then $4 θ + 50^{\circ} = 90^{\circ}$ . Solving this linear equation gives $4 θ = 40^{\circ}$ , so $θ = 10^{\circ}$ . This property allows us to solve complex trigonometric equations quickly.

Complementary angles sum to $90^{\circ}$ , making the sine of one equal to the cosine of the other.
Equate the sum of the two angle arguments to $90^{\circ}$ when sine equals cosine.
Solve the resulting algebraic equation to calculate the unknown variable or acute angle.
Expressions like $\cos (\frac{θ}{3}) = \sin 20^{\circ}$ simplify directly by setting $\frac{θ}{3} + 20^{\circ} = 90^{\circ}$ .

Worked example. For (a), set the sum of arguments equal to

90^{\circ}

α + 52^{\circ} = 90^{\circ}

, so

α = 90^{\circ} - 52^{\circ} = 38^{\circ}

. For (b), equate the sum to

90^{\circ}

5 α + 35^{\circ} = 90^{\circ}

, which gives

5 α = 55^{\circ}

, so

α = 11^{\circ}

. For (c), write

\frac{α}{2} + 70^{\circ} = 90^{\circ}

, so

\frac{α}{2} = 20^{\circ}

, yielding

α = 40^{\circ}

. Answer: (a) $α = 38^{\circ}$ , (b) $α = 11^{\circ}$ , (c) $α = 40^{\circ}$ .

Check question: Find the value of $θ$ if $\sin (4 θ) = \cos 18^{\circ}$ , where $4 θ$ is an acute angle.

${4.5}^{\circ}$
$18^{\circ}$
$27^{\circ}$
$72^{\circ}$

Show answer

$18^{\circ}$ . Since sine equals cosine for complementary angles, their arguments must sum to $90^{\circ}$ . We set up the equation $4 θ + 18^{\circ} = 90^{\circ}$ . Subtracting $18^{\circ}$ from both sides gives $4 θ = 72^{\circ}$ . Dividing by 4 gives $θ = \frac{72^{\circ}}{4} = 18^{\circ}$ .

Slide 2 · Applying Complementary Ratios in TrianglesPractice · 22 min

Figure 2: A right-angled triangle labelled with side lengths of 8, 15, and 17 centimetres and two acute angles.

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Both slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 10 lessons in this sub-strand

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Solving Trigonometric Equations with Complementary Angles

By the end of this lesson, learners should be able to

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 11 lessons in Trigonometry 1