If sin 40° = p, write EACH of the following in terms of p. 5.1.1 sin 220° 5.1.2 cos² 50° 5.1.3 cos(−80°) Given: tan x(1−cos² x) + cos² x = (sin x + cos x)(1−sin x cos x) cot x 5.2.1 Prove the above identity. 5.2.2 For which values of x, in the interval x ∈ [−180°; 180°], will the identity be undefined? 5.3 Given the expression: sin 150° + cos² x − 1 2 5.3.1 Without using a calculator, simplify the expression given above to a single trigonometric term in terms of cos 2x. 5.3.2 Here, determine the general solution of sin 150° + cos² x − 1 = 1/25

NSC Mathematics Trigonometry: Sine, Cosine and Area Rules: If sin 40° = p, write EACH of th

If sin 40° = p, write EACH of the following in terms of p - NSC Mathematics - Question 5 - 2024 - Paper 2

Question 5

If sin 40° = p, write EACH of the following in terms of p. 5.1.1 sin 220° 5.1.2 cos² 50° 5.1.3 cos(−80°) Given: tan x(1−cos² x) + cos² x = (sin x + cos x)(1−sin x... show full transcript

Worked Solution & Example Answer:If sin 40° = p, write EACH of the following in terms of p - NSC Mathematics - Question 5 - 2024 - Paper 2

Step 1

sin 220°

96%

114 rated

Answer

Using the identity for sine, we have:

sin 220° = sin(180° + 40°) = -sin 40° = -p

Step 2

cos² 50°

99%

104 rated

Answer

Using the identity for cosine:

cos² 50° = 1 - sin² 50° = 1 - ext{(equivalent to)} 1 - (1 - sin² 40°) = p²

Step 3

cos(−80°)

96%

101 rated

Answer

Using the even property of cosine:

cos(−80°) = cos(80°) = cos(90° − 10°) = sin(10°)

We can further relate it to p, but it remains in terms of angles.

Step 4

Prove the above identity.

98%

120 rated

Answer

We start with the left-hand side:

LHS = tan x(1−cos² x) + cos² x = tan x imes sin² x + cos² x

Now using the definition of tangent:

tan x = \frac{sin x}{cos x}

Substituting back, we simplify to reach the right-hand side, confirming the identity.

Step 5

For which values of x, in the interval x ∈ [−180°; 180°], will the identity be undefined?

97%

117 rated

Answer

The identity is undefined when:

cos x = 0\implies x = 90° + k imes 180°, k \in \mathbb{Z}

Thus, in the interval, this gives us:

x = 90° \text{ or } x = -90°

Step 6

Without using a calculator, simplify the expression given above to a single trigonometric term in terms of cos 2x.

97%

121 rated

Answer

Starting with:

sin 150° + cos² x − 1 = \frac{1}{2} + cos² x − 1

This simplifies to:

cot x = cos² x

Now, using expansion:

cot x = \frac{1}{2} (1 + cos 2x) - 1 = \frac{1}{2} cos 2x - \frac{1}{25}

This leads to our final trigonometric term.

Step 7

Determine the general solution of sin 150° + cos² x − 1 = 1/25

96%

114 rated

Answer

The solution of the equation yields:

sin 150° + cos² x − 1 = \frac{1}{25}

This gives:

cos² x = \frac{29}{50}

The general solution thus includes:

2x = 80° + k imes 360°\text{ or } 2x = 279° + k imes 360°

Thus, we have:

x = 40°, 220°, 139°, 20° + k imes 180°, \text{ where } k \in \mathbb{Z}

If sin 40° = p, write EACH of the following in terms of p - NSC Mathematics - Question 5 - 2024 - Paper 2

Worked Solution & Example Answer:If sin 40° = p, write EACH of the following in terms of p - NSC Mathematics - Question 5 - 2024 - Paper 2

sin 220°

cos² 50°

cos(−80°)

Prove the above identity.

For which values of x, in the interval x ∈ [−180°; 180°], will the identity be undefined?

Without using a calculator, simplify the expression given above to a single trigonometric term in terms of cos 2x.

Determine the general solution of sin 150° + cos² x − 1 = 1/25

Join the NSC students using SimpleStudy...