1. §4.3: Right Triangle Trigonometry
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§4.3: Right Triangle Trigonometry

2. Right Triangle Trigonometry
Section 4.3
Precalculus PreAP/Dual, Revised ©2017
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§4.3: Right Triangle Trigonometry

3. Background Information
Trigonometry
Comes from Greek word – Trigonon, which means 3 angles
“Metry” means measure in Greek
Trigonometry Ratios
Main: Sine, Cosine, Tangent
Reciprocal: Secant, Cosecant, Cotangent
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§4.3: Right Triangle Trigonometry

4. §4.3: Right Triangle Trigonometry
Right Triangles
hypotenuse
opposite
adjacent
Consider a right triangle, one of whose acute angles is 𝜽
The three sides of a triangle are hypotenuse, opposite, and adjacent side of 𝜽
To determine what is the opposite side, look at 𝜽 and extend the line to determine the opposite
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§4.3: Right Triangle Trigonometry

5. Steps in Solving for Triangles
Solve for 𝒙, using Pythagorean Theorem
Determine the missing sides by identifying 𝜽
Use Trigonometry Functions to find what’s needed
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§4.3: Right Triangle Trigonometry

6. §4.3: Right Triangle Trigonometry
Trigonometric Ratios
𝐒𝐢𝐧𝐞 𝜽= 𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝐂𝐨𝐬𝐞𝐜𝐚𝐧𝐭 𝜽= 𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝐒𝐈𝐍 𝐂𝐒𝐂 𝐂𝐨𝐬𝐢𝐧𝐞 𝜽= 𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝐒𝐞𝐜𝐚𝐧𝐭 𝜽= 𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝐂𝐎𝐒 𝐒𝐄𝐂 𝐓𝐚𝐧𝐠𝐞𝐧𝐭 𝜽= 𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝐂𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭 𝜽= 𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝐓𝐀𝐍 𝐂𝐎𝐓
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§4.3: Right Triangle Trigonometry

7. §4.3: Right Triangle Trigonometry
Co-Functions
𝐒𝐢𝐧𝐞 𝜽= 𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝐂𝐨𝐬𝐢𝐧𝐞 𝜽= 𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝐒𝐈𝐍 𝐂𝐎𝐒 𝐂𝐨𝐬𝐞𝐜𝐚𝐧𝐭 𝜽= 𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝐒𝐞𝐜𝐚𝐧𝐭 𝜽= 𝑯𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝐂𝐎𝐒 𝐒𝐄𝐂 𝐓𝐚𝐧𝐠𝐞𝐧𝐭 𝜽= 𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝐂𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭 𝜽= 𝑨𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝑶𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝐓𝐀𝐍 𝐂𝐎𝐓
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§4.3: Right Triangle Trigonometry

8. §4.3: Right Triangle Trigonometry
Example 1
Solve for 𝒙 and determine all trig functions of 𝜽
HYPOTENUSE 13 12
Opposite 5 x
ADJACENT
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§4.3: Right Triangle Trigonometry

9. §4.3: Right Triangle Trigonometry
Example 1
Solve for 𝒙 and determine all trig functions of 𝜽 13 12 5
HYPOTENUSE
Opposite 𝒙
ADJACENT
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§4.3: Right Triangle Trigonometry

10. §4.3: Right Triangle Trigonometry
Example 2
Solve for 𝒙 and determine all trig functions of 𝜽 𝟓 𝒙
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§4.3: Right Triangle Trigonometry

11. §4.3: Right Triangle Trigonometry
Your Turn
Solve for 𝒙 and determine all trig functions of 𝜽 𝟐 𝒙 1
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§4.3: Right Triangle Trigonometry

12. Special Right Triangles
Drawing is not to scale 𝟒𝟓°, 𝟒𝟓°, and 𝟗𝟎° 𝟑𝟎°, 𝟔𝟎°, and 𝟗𝟎°
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§4.3: Right Triangle Trigonometry

13. §4.3: Right Triangle Trigonometry
Example 3
Find the exact value of 𝒙 without using a calculator
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§4.3: Right Triangle Trigonometry

14. §4.3: Right Triangle Trigonometry
Example 4
Find the exact value of 𝒙 without using a calculator
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§4.3: Right Triangle Trigonometry

15. §4.3: Right Triangle Trigonometry
Your Turn
Find the exact value of 𝒙 without using a calculator
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§4.3: Right Triangle Trigonometry

16. §4.3: Right Triangle Trigonometry
Example 5
Given 𝐜𝐨𝐬 𝜽= 𝟓 𝟔 , identify the third side and solve for 𝐬𝐞𝐜 𝜽 and 𝐜𝐨𝐭 𝜽
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§4.3: Right Triangle Trigonometry

17. §4.3: Right Triangle Trigonometry
Example 5
Given 𝐜𝐨𝐬 𝜽= 𝟓 𝟔 , identify the third side and solve for 𝐬𝐞𝐜 𝜽 and 𝐜𝐨𝐭 𝜽
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§4.3: Right Triangle Trigonometry

18. §4.3: Right Triangle Trigonometry
Example 6
Given 𝐜𝐨𝐭 𝜽=𝟓, identify the third side and solve for 𝐭𝐚𝐧 𝜽 and 𝐜𝐨𝐭 𝟗𝟎°−𝜽
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§4.3: Right Triangle Trigonometry

19. §4.3: Right Triangle Trigonometry
Your Turn
Given 𝐜𝐬𝐜 𝜽=𝟗, identify the third side and solve for 𝐜𝐨𝐬 𝜽 and 𝐜𝐨𝐭 𝜽
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§4.3: Right Triangle Trigonometry

20. §4.3: Right Triangle Trigonometry
Assignment
Page 286 5, 8, odd, odd, all
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§4.3: Right Triangle Trigonometry