In a right-angled triangle PQR with R = 90°, if P = (5)/(13), what is the value of Q?

(5)/(13). Since the sum of angles in a triangle is 180° and R = 90°, angles P and Q add up to 90°. They are complementary angles, which means Q = P = (5)/(13).

Grade 10 · Core Mathematics · Lesson 4 of 11

Connecting Sine and Cosine in Complementary Angles

A Grade 10 Core Mathematics CBE lesson from the sub-strand Trigonometry 1. In it, learners deduce that the sine of an acute angle equals the cosine of its complementary angle in a right-angled triangle.

Strand Measurements and GeometrySub-strand Trigonometry 1Lesson code 10CM-24-L440 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

Deduce that the sine of an acute angle equals the cosine of its complementary angle in a right-angled triangle.
Express sine and cosine ratios of complementary angles using side lengths.
Solve algebraic equations involving sines and cosines of complementary angles.

Key vocabulary

Complementary angles: Two angles whose sum is equal to ninety degrees.; Angles measuring thirty degrees and sixty degrees are complementary angles.
Sine: Ratio of the length of the opposite side to the hypotenuse.; The sine of thirty degrees in this triangle equals zero point five.
Cosine: Ratio of the length of the adjacent side to the hypotenuse.; The cosine of sixty degrees equals zero point five in right triangles.
Acute angle: An angle that measures strictly less than ninety degrees.; An angle of twenty-five degrees is classified as an acute angle.

Lesson preview

Slide 1 · Discovering Relationships in Complementary AnglesOpening · 18 min

Figure 1: A right-angled triangle with side lengths of 6 cm, 8 cm, and 10 cm, alongside a table comparing the trigonometric ratios of its complementary angles.

Two acute angles are complementary when their sum is $90^{\circ}$ . In a right-angled triangle with acute angles $θ$ and $90^{\circ} - θ$ , the side opposite one angle serves as the adjacent side for the complement. Consequently, the sine ratio of $θ$ equals $\cos (90^{\circ} - θ)$ , while the cosine ratio of $θ$ equals $\sin (90^{\circ} - θ)$ .

Complementary angles are two angles that sum to exactly $90^{\circ}$ .
In a right-angled triangle, the two acute angles are always complementary to each other.
The side opposite to angle $θ$ acts as the adjacent side to angle $90^{\circ} - θ$ .
The sine of an acute angle equals the cosine of its complementary angle.

Worked example. Identify angle

X = θ

and angle

Y = 90^{\circ} - θ

, noting that

X + Y = 90^{\circ}

. Calculate

\sin X = \frac{opposite}{hypotenuse} = \frac{6}{10} = 0.6

. Calculate

\cos X = \frac{adjacent}{hypotenuse} = \frac{8}{10} = 0.8

. Calculate

\sin Y = \frac{opposite}{hypotenuse} = \frac{8}{10} = 0.8

and

\cos Y = \frac{adjacent}{hypotenuse} = \frac{6}{10} = 0.6

. Observe that

\sin X = \cos Y = 0.6

and

\cos X = \sin Y = 0.8

. Answer: $\sin X = \cos Y = 0.6$ and $\cos X = \sin Y = 0.8$ , confirming $\sin θ = \cos (90^{\circ} - θ)$ ..

Check question: In a right-angled triangle $P Q R$ with $∠ R = 90^{\circ}$ , if $\sin P = \frac{5}{13}$ , what is the value of $\cos Q$ ?

$\frac{5}{13}$
$\frac{12}{13}$
$\frac{5}{12}$
$\frac{13}{5}$

Show answer

$\frac{5}{13}$ . Since the sum of angles in a triangle is $180^{\circ}$ and $∠ R = 90^{\circ}$ , angles $P$ and $Q$ add up to $90^{\circ}$ . They are complementary angles, which means $\cos Q = \sin P = \frac{5}{13}$ .

Slide 2 · Solving Equations Using Complementary IdentitiesPractice · 22 min

Figure 2: A step-by-step flowchart demonstrating how to solve a trigonometric equation by converting it into a linear equation using the complementary angle identity.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

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

See the school projector WhatsApp us

Connecting Sine and Cosine in 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