example: find all three trig ratios in relation to the angle given.
you try! find the exact values of the three trigonometric functions of $\theta$.
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practice/homework: find the exact values of the three trigonometric functions of $\theta$.
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4. in the triangle below, compare and contrast the trig ratios for angle $x$ and angle $z$. can you make any connections between them?

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Accepted Answer

{ "quality": "clear", "question_count": 6, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Trigonometric Ratios", "Right Triangle Trigonometry", "Pythagorean Theorem Application" ], "new_concepts": [], "current_concepts": [ "Trigonometric Ratios", "Right Triangle Trigonometry", "Pythagorean Theorem Application" ] } ## Find missing sides using Pythagorean theorem $$ \begin{aligned} &\text{You TRY! 1: } \text{Adjacent} = \sqrt{29^2 - 20^2} = \sqrt{841 - 400} = \sqrt{441} = 21\ &\text{You TRY! 2: } \text{Adjacent} = \sqrt{13^2 - 4^2} = \sqrt{169 - 16} = \sqrt{153} = 3\sqrt{17}\ &\text{Practice 1: } \text{Opposite} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3\ &\text{Practice 2: } \text{Opposite} = 7,\ \text{Adjacent} = 7\sqrt{3},\ \text{Hypotenuse} = 14\ &\text{Practice 3: } \text{Opposite} = \sqrt{34^2 - 30^2} = \sqrt{1156 - 900} = \sqrt{256} = 16 \end{aligned} $$ ## Calculate trigonometric ratios for each triangle $$ \begin{aligned} &\text{You TRY! 1: } \sin\theta = \frac{20}{29},\ \cos\theta = \frac{21}{29},\ \tan\theta = \frac{20}{21}\ &\text{You TRY! 2: } \sin\theta = \frac{4}{13},\ \cos\theta = \frac{3\sqrt{17}}{13},\ \tan\theta = \frac{4}{3\sqrt{17}} = \frac{4\sqrt{17}}{51}\ &\text{Practice 1: } \sin\theta = \frac{3}{5},\ \cos\theta = \frac{4}{5},\ \tan\theta = \frac{3}{4}\ &\text{Practice 2: } \sin\theta = \frac{7}{14} = \frac{1}{2},\ \cos\theta = \frac{7\sqrt{3}}{14} = \frac{\sqrt{3}}{2},\ \tan\theta = \frac{7}{7\sqrt{3}} = \frac{\sqrt{3}}{3}\ &\text{Practice 3: } \sin\theta = \frac{16}{34} = \frac{8}{17},\ \cos\theta = \frac{30}{34} = \frac{15}{17},\ \tan\theta = \frac{16}{30} = \frac{8}{15} \end{aligned} $$ ## Compare and contrast ratios for Question 4 $$ \begin{aligned} &\text{For angle } X: \sin X = \frac{\sqrt{11}}{6},\ \cos X = \frac{5}{6},\ \tan X = \frac{\sqrt{11}}{5}\ &\text{For angle } Z: \sin Z = \frac{5}{6},\ \cos Z = \frac{\sqrt{11}}{6},\ \tan Z = \frac{5}{\sqrt{11}} = \frac{5\sqrt{11}}{11}\ &\text{Connections: } \sin X = \cos Z,\ \cos X = \sin Z,\ \tan X = \frac{1}{\tan Z} \end{aligned} $$ | No. | Problem | Answer | |---|---|---| | You TRY! 1 | Find the exact values of the three trigonometric functions of $\theta$. | $\sin\theta = \frac{20}{29}$, $\cos\theta = \frac{21}{29}$, $\tan\theta = \frac{20}{21}$ | | You TRY! 2 | Find the exact values of the three trigonometric functions of $\theta$. | $\sin\theta = \frac{4}{13}$, $\cos\theta = \frac{3\sqrt{17}}{13}$, $\tan\theta = \frac{4\sqrt{17}}{51}$ | | Practice 1 | Find the exact values of the three trigonometric functions of $\theta$. | $\sin\theta = \frac{3}{5}$, $\cos\theta = \frac{4}{5}$, $\tan\theta = \frac{3}{4}$ | | Practice 2 | Find the exact values of the three trigonometric functions of $\theta$. | $\sin\theta = \frac{1}{2}$, $\cos\theta = \frac{\sqrt{3}}{2}$, $\tan\theta = \frac{\sqrt{3}}{3}$ | | Practice 3 | Find the exact values of the three trigonometric functions of $\theta$. | $\sin\theta = \frac{8}{17}$, $\cos\theta = \frac{15}{17}$, $\tan\theta = \frac{8}{15}$ | | Question 4 | Compare and contrast the trig ratios for angle $X$ and angle $Z$. | $\sin X = \cos Z = \frac{\sqrt{11}}{6}$; $\cos X = \sin Z = \frac{5}{6}$; $\tan X = \frac{1}{\tan Z} = \frac{\sqrt{11}}{5}$. They are cofunctions because $X$ and $Z$ are complementary angles. | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Right Triangle Trigonometry" ] }

No.	Problem	Answer
You TRY! 2	Find the exact values of the three trigonometric functions of \(\theta\).	\(\sin\theta = \frac{4}{13}\), \(\cos\theta = \frac{3\sqrt{17}}{13}\), \(\tan\theta = \frac{4\sqrt{17}}{51}\)
Practice 1	Find the exact values of the three trigonometric functions of \(\theta\).	\(\sin\theta = \frac{3}{5}\), \(\cos\theta = \frac{4}{5}\), \(\tan\theta = \frac{3}{4}\)
Practice 2	Find the exact values of the three trigonometric functions of \(\theta\).	\(\sin\theta = \frac{1}{2}\), \(\cos\theta = \frac{\sqrt{3}}{2}\), \(\tan\theta = \frac{\sqrt{3}}{3}\)
Practice 3	Find the exact values of the three trigonometric functions of \(\theta\).	\(\sin\theta = \frac{8}{17}\), \(\cos\theta = \frac{15}{17}\), \(\tan\theta = \frac{8}{15}\)
Question 4	Compare and contrast the trig ratios for angle \(X\) and angle \(Z\).	\(\sin X = \cos Z = \frac{\sqrt{11}}{6}\); \(\cos X = \sin Z = \frac{5}{6}\); \(\tan X = \frac{1}{\tan Z} = \frac{\sqrt{11}}{5}\). They are cofunctions because \(X\) and \(Z\) are complementary angles.

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Question

Explanation:

Find missing sides using Pythagorean theorem

Calculate trigonometric ratios for each triangle

Compare and contrast ratios for Question 4

Answer: