6.6 The Sine and Cosine Ratios Tỉ số lượng giác sin và cos

The ladder, the wall and the ground form a right triangle. The ladder is the hypotenuse, $6$ m. Let $h$ be the height it reaches (opposite the $72^\circ$ angle) and $d$ the distance of its foot from the building (adjacent):

Answer: a. about $5.7$ m; b. about $1.9$ m.

The wire, the tower and the ground form a right triangle. The wire $w$ is the hypotenuse and $24$ m is the leg adjacent to the $55^\circ$ angle:

One extra meter at each end adds $2$ m: $41.84 + 2 = 43.84.$

Answer: about $43.8$ m of wire.

At the angle $x^\circ,$ the opposite leg is $9$ and the hypotenuse is $41,$ so we use the sine:

This lies between $\sin 12^\circ \approx 0.2079$ and $\sin 13^\circ \approx 0.2250$ and is closer to $0.2250.$

Answer: $x \approx 13.$

At the angle $x^\circ,$ the opposite leg is $4$ and the hypotenuse is $9,$ so we use the sine:

This lies between $\sin 26^\circ \approx 0.4384$ and $\sin 27^\circ \approx 0.4540$ and is closer to $0.4384.$

Answer: $x \approx 26.$

Let $\triangle TLR$ have $TL = TR = 16$ and base angles $\angle L = \angle R = 42^\circ.$ Draw the altitude $\overline{TM}$ to the base. The right triangles $TLM$ and $TRM$ are congruent by HL, so $LM = MR.$

In $\triangle TLM,$ the hypotenuse is $TL = 16$ and $LM$ is adjacent to the $42^\circ$ angle:

Answer: the base is about $23.8$ cm.

Let $\triangle TLR$ have $TL = TR,$ $m\angle T = 48^\circ$ and $LR = 30.$ Draw the altitude $\overline{TM}.$ The right triangles $TLM$ and $TRM$ are congruent by HL, so $LM = MR = 15$ and $m\angle LTM = m\angle RTM = 24^\circ.$

In $\triangle TLM,$ $LM = 15$ is opposite the $24^\circ$ angle and $TL$ is the hypotenuse:

Answer: each leg is about $36.9$ cm.

$\triangle ABD$ has a right angle at $D$ and hypotenuse $AB = 40$; $DB$ is opposite the $55^\circ$ angle:

$\triangle DCB$ has a right angle at $C$ and hypotenuse $DB$; $x = DC$ is adjacent to the $32^\circ$ angle:

Answer: $x \approx 27.8.$

The altitude $h$ is a leg of both right triangles. In the right one, $25$ is the hypotenuse and $h$ is opposite the $44^\circ$ angle:

In the left one, $x$ is the hypotenuse and $h$ is opposite the $50^\circ$ angle:

Answer: $x \approx 22.7.$