Question

Please help with these 3 questions-

Answer 1

See attachments.

Explanation:

Trigonometric ratios are mathematical expressions defining the relationships between the angles and sides of a right-angled triangle.

The primary trigonometric ratios are:

`\boxed{\begin{array}{l}\underline{\sf Trigonometric\;ratios}\\\\\sf \sin(\theta)=(O)/(H)\qquad\cos(\theta)=(A)/(H)\qquad\tan(\theta)=(O)/(A)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{O is the side opposite the angle.}\\\phantom{ww}\bullet\;\textsf{A is the side adjacent the angle.}\\\phantom{ww}\bullet\;\textsf{H is the hypotenuse (the side opposite the right angle).}\\\\\end{array}}`

`\hrulefill`

Question 4

Given:

`\sin(A)=(1)/(2)`

The provided sine ratio implies that the side opposite angle A is one unit in length, and the hypotenuse is 2 units.

In a right triangle, the side opposite the right angle is the hypotenuse.

In right triangle ABC, let ∠C = 90°, which makes side c the hypotenuse, and sides a and b the legs. So, a = 1 and c = 2.

To find the length of leg b (the side adjacent to angle A), substitute a = 1 and c = 2 into the Pythagorean Theorem:

`\begin{aligned}a^2+b^2&=c^2\\1^2+b^2&=2^2\\1+b^2&=4\\b^2&=3\\b&=√(3)\end{aligned}`

Therefore, the length of leg b is b = √3.

We can find the measure of angle A by solving using the given ratio:

`\sin(A)=(1)/(2)\implies A=\arcsin\left((1)/(2)\right)=30^(\circ)`

Since the interior angles of a triangle sum to 180°, then:

`B=180^(\circ)-A-C`

`B=180^(\circ)-30^(\circ)-90^(\circ)`

`B=60^(\circ)`

Therefore:

• a = 1
• b = √3
• c = 2
• A = 30°
• B = 60°
• C = 90°

These values allow us to sketch the right triangle based on the determined lengths and angles (see attachment 1).

`\hrulefill`

Question 5

Given:

`\cos(B)=\frac35`

The provided cosine ratio implies that the side adjacent angle B is 3 units in length, and the hypotenuse is 5 units.

In a right triangle, the side opposite the right angle is the hypotenuse.

In right triangle ABC, let ∠C = 90°, which makes side c the hypotenuse, and sides a and b the legs. So, a = 3 and c = 5.

To find the length of leg b (the side adjacent to angle A), substitute a = 3 and c = 5 into the Pythagorean Theorem:

`\begin{aligned}a^2+b^2&=c^2\\3^2+b^2&=5^2\\9+b^2&=25\\b^2&=16\\b&=4\end{aligned}`

Therefore, the length of leg b is b = 4.

We can find the measure of angle B by solving using the given ratio:

`\cos(B)=(3)/(5)\implies B=\arccos\left((3)/(5)\right)=53.1^(\circ)`

Since the interior angles of a triangle sum to 180°, then:

`A=180^(\circ)-B-C`

`A=180^(\circ)-53.1^(\circ)-90^(\circ)`

`A=36.9^(\circ)`

Therefore:

• a = 3
• b = 4
• c = 5
• A = 36.9°
• B = 53.1°
• C = 90°

These values allow us to sketch the right triangle based on the determined lengths and angles (see attachment 2).

`\hrulefill`

Question 6

Given:

`\tan(B)=\frac67`

The provided tangent ratio implies that the side opposite angle B is 6 units in length, and the side adjacent angle B is 7 units.

In a right triangle, the side opposite the right angle is the hypotenuse.

In right triangle ABC, let ∠C = 90°, which makes side c the hypotenuse, and sides a and b the legs. So, a = 7 and b = 6.

To find the length of the hypotenuse, substitute a = 7 and b = 6 into the Pythagorean Theorem, and solve for c:

`\begin{aligned}a^2+b^2&=c^2\\7^2+6^2&=c^2\\49+36&=c^2\\85&=c^2\\c&=√(85)\end{aligned}`

Therefore, the length of the hypotenuse is c = √(85).

We can find the measure of angle B by solving using the given ratio:

`\tan(B)=\frac67\implies B=\arctan\left(\frac67\right)=40.6^(\circ)`

Since the interior angles of a triangle sum to 180°, then:

`A=180^(\circ)-B-C`

`A=180^(\circ)-40.6^(\circ)-90^(\circ)`

`A=49.4^(\circ)`

Therefore:

• a = 7
• b = 6
• c = √(85)
• A = 49.4°
• B = 40.6°
• C = 90°

These values allow us to sketch the right triangle based on the determined lengths and angles (see attachment 3).