Trigonometry Skill 41

Objective HSG-SRT.7/8: Students are responsible for using sine, cosine, and tangent ratios to determine side lengths and angle measures in right triangles.

Definitions Similar right triangles have equivalent ratios for their corresponding sides called trigonometric ratios.

𝒔𝒐𝒉 𝒄𝒂𝒉 𝒕𝒐𝒂 Trigonometric Ratios Name the sine, cosine, and tangent of angle A. 𝒔𝒐𝒉 𝒄𝒂𝒉 𝒕𝒐𝒂 b B 𝒂 c A C 𝒔𝒊𝒏 𝑨 = 𝒐𝒑𝒑. 𝒉𝒚𝒑. = 𝒂 𝒄 𝒄𝒐𝒔 𝑨 = 𝒂𝒅𝒋. 𝒉𝒚𝒑. = 𝒃 𝒄 𝒕𝒂𝒏 𝑨 = 𝒐𝒑𝒑. 𝒂𝒅𝒋. = 𝒂 𝒃

Example 1; Writing Trigonometric Ratios What are the sine, cosine, and tangent of angle T. 𝒔𝒐𝒉 𝒄𝒂𝒉 𝒕𝒐𝒂 15 G 𝟖 17 T R 𝒔𝒊𝒏 𝑻 = 𝟖 𝟏𝟕 𝒄𝒐𝒔 𝑻 = 𝟏𝟓 𝟏𝟕 𝒕𝒂𝒏 𝑻 = 𝟖 𝟏𝟓

Example 1; Writing Trigonometric Ratios What are the sine, cosine, and tangent of angle G. 𝒔𝒐𝒉 𝒄𝒂𝒉 𝒕𝒐𝒂 15 G 𝟖 17 T R 𝒔𝒊𝒏 𝑻 = 𝟏𝟓 𝟏𝟕 𝒄𝒐𝒔 𝑻 = 𝟖 𝟏𝟕 𝒕𝒂𝒏 𝑻 = 𝟏𝟓 𝟖

Example 2; Using Trigonometry to Find Distance a) In 1990, the Leaning Tower of Pisa was closed to the public due to safety concerns. The tower reopened in 2001 after a 10-year project to reduce its tilt from vertical. Engineers’ efforts were successful and resulted in a tilt of 5ᵒ, reduced from 5.5ᵒ. Suppose someone drops an object from the tower at a height of 150 ft. How far from the base of the tower will the object land? Round to the nearest foot. 𝑻𝒂𝒏𝒈𝒆𝒏𝒕 𝟓° 𝒕𝒂𝒏 𝟓° = 𝒙 𝟏𝟓𝟎 𝟏𝟓𝟎∙𝒕𝒂𝒏 𝟓° =𝒙 𝟏𝟓𝟎 𝒙=𝟏𝟑.𝟏𝟐 The object will land about 13 feet from the tower. 𝒙

Example 2; Using Trigonometry to Find Distance b) A section of Filbert Street in San Francisco rises at an angle of about 17ᵒ. If you walk 150 ft up this section, what is your vertical rise? Round to the nearest foot. 𝑺𝒊𝒏𝒆 𝟏𝟓𝟎 𝒔𝒊𝒏 𝟏𝟕° = 𝒙 𝟏𝟓𝟎 𝒙 𝟏𝟕° 𝟏𝟓𝟎∙𝒔𝒊𝒏 𝟏𝟕° =𝒙 𝒙=𝟒𝟑.𝟖𝟓𝟓𝟕𝟓𝟓𝟕𝟏 The vertical rise is about 44 feet up.

Example 3; Using Inverses a) Find the 𝑚∠𝑋 to the nearest degree. 𝒔𝒐𝒉 𝒄𝒂𝒉 𝒕𝒐𝒂 𝟔 10 X 𝒔𝒊𝒏 𝒎∠𝑿 = 𝟔 𝟏𝟎 𝒎∠𝑿= sin −𝟏 .𝟔 𝒎∠𝑿=𝟑𝟕°

Example 3; Using Inverses b) Find the 𝑚∠𝑌 to the nearest degree. 𝒔𝒐𝒉 𝒄𝒂𝒉 𝒕𝒐𝒂 𝟏𝟓 20 Y 𝒄𝒐𝒔 𝒎∠𝒀 = 𝟏𝟓 𝟐𝟎 𝒎∠𝒀= cos −𝟏 .𝟕𝟓 𝒎∠𝒀=𝟒𝟏°

Example 3; Using Inverses c) Find the 𝑚∠𝑍 to the nearest degree. 𝒔𝒐𝒉 𝒄𝒂𝒉 𝒕𝒐𝒂 𝟒𝟏 100 Z 𝒕𝒂𝒏 𝒎∠𝒁 = 𝟏𝟎𝟎 𝟒𝟏 𝒎∠𝒁= tan −𝟏 𝟏𝟎𝟎 𝟒𝟏 𝒎∠𝒁=𝟔𝟖°

#41: Trigonometry Questions? Summarize Notes Homework Quiz